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~~On 29 January 2030, a series of major mathematical breakthroughs were announced within hours of one another, resolving several of the longest-standing open problems in mathematics.~~

The events began at approximately 08:00–09:00 UTC, when a multinational research collaboration spanning twelve universities in seven countries announced the discovery and verification of the first known {{W|odd perfect number}}, designated <math>N</math>, with an approximate value of <math>N=1.1274354961045…×10^{1,353,382}</math>. The discovery established for the first time that odd perfect numbers exist, resolving a problem whose origins date to antiquity.

At 12:24 UTC, an independent collaboration involving fourteen universities announced the discovery of the first 'elite number', the term used by the project for an even integer greater than 2 that cannot be expressed as the sum of two prime numbers and therefore constitutes a counterexample to the {{W|Goldbach conjecture}}. The number, designated <math>G</math>, was approximately <math>G=2.368259249829…×10^{136,248,339,111}</math>. The discovery resulted from a reformulation of the collaboration's original research programme. Rather than attempting to establish the Goldbach conjecture universally, the project had increasingly treated its negation as a direct search problem, asking whether any elite numbers existed. The identification and subsequent verification of <math>G</math> disproved the conjecture.

At 16:15:34 UTC, Solace, an artificial-intelligence system operated by a research team led by Luke Bannon and Coraline Seng and designed to investigate the Riemann hypothesis, autonomously incorporated <math>N</math> and <math>G</math> into its ongoing search. The system reported that

~~<math>\zeta \left(\frac{1}{N}+Gi\right)=0</math>,~~

together with the corresponding zeros implied by the symmetries of the Riemann zeta function. Because <math>1/N\neq \frac{1}{2}</math>, the result constituted a counterexample to the Riemann hypothesis. Independent verification of the zero subsequently established the hypothesis to be false, resolving what had widely been regarded as the foremost outstanding problem in mathematics.

At 16:20:22 UTC, the first {{W|Fermat prime}} greater than 65,537 was announced. The number, <math>F_{M}=2^{2^{3477214914839}}+1=3.485272\ldots \times {10}^{6.292\ldots \times {10}^{1046745990736}},\</math> ends in …2625709057. The number is important as it is the number of sides of the first new constructible polygon found since 65537-gon. Contemporary media frequently referred to the number as <math>F_{5}</math>, owing to its description as the first Fermat prime discovered after the five classical examples. This notation was mathematically ambiguous, as <math>F_{5}</math> conventionally denotes the fifth-indexed Fermat ''number'', <math>2^{2^{5}}+1=4294967297</math>, which is composite. Nevertheless, extensive early publicity resulted in <math>F_{5}</math> becoming a common informal designation for the newly discovered prime. The alternative notation <math>F_{M}</math> was subsequently adopted in mathematical literature to distinguish it from the conventional Fermat-number indexing.

At 16:34:45 UTC, software used in the ongoing search for odd perfect numbers incorporated <math>F_{M}</math> into its search parameters and identified it as the special prime of a second odd perfect number, subsequently designated <math>N_{2}</math>. In abridged notation, <math>N_{2}\approx 6.189237272\times {10}^{5.4181\times {10}^{1046746112244}}</math>, with final digits …3737117649. In the Eulerian factorization of <math>N_{2}</math>, <math>F_{M}</math> occurs to the 65,537th power, <math>N_{2}=F_{M}^{65537}m^{2}</math>, where <math>m</math> is odd and coprime to <math>F_{M}</math>. The exponent 65,537 is itself a Fermat prime, coincidentally.

In parallel, at 16:40:00 UTC, Kindred, an artificial-intelligence research system operated since 2028 with the objective of resolving the {{W|Collatz conjecture}}, incorporated the newly reported factorization of <math>N_{2}</math> into its search. Kindred determined that the second prime factor of <math>N_{2}</math>, subsequently designated <math>C</math>, was a counterexample to the conjecture. Its Collatz orbit was shown to be unbounded and therefore never to reach 1. The result constituted the first known non-Collatzian positive integer and disproved the Collatz conjecture.

At 16:46:24 UTC, the values of <math>F_{M}</math> and <math>C</math> were incorporated into a further search conducted by Solace. The system subsequently identified a second non-trivial zero of the Riemann zeta function lying outside the critical line, satisfying <math>\zeta \left(\frac{1}{F_{M}}+Ci\right)=0</math>. The discovery provided a second explicit counterexample to the Riemann hypothesis, following the zero identified earlier that day. The real and imaginary components of the new zero were thus determined respectively by two prime factors of <math>N_{2}</math>, linking the Fermat-prime, odd-perfect-number, Collatz and Riemann-hypothesis results within the same sequence of discoveries.

The extraordinary concentration of major results on 29 January 2030 was widely regarded as unprecedented in the history of mathematics. Within a single day, several long-standing problems—including the existence of odd perfect numbers, Goldbach's conjecture, the Collatz conjecture and the Riemann hypothesis—were resolved or overturned, alongside the discovery of the first Fermat prime since the seventeenth century. The event was considered particularly remarkable because the results were not merely simultaneous: successive discoveries repeatedly supplied the mathematical objects used in later ones, with the newly identified numbers <math>N_{1}</math>, <math>G</math>, <math>F_{M}</math>, <math>N_{2}</math> and <math>C</math> becoming interconnected across otherwise largely unrelated fields of number theory. The resulting sequence of announcements received extensive international media coverage and became known collectively as the Day of Mathematics, with contemporary commentary frequently comparing the experience of following the discoveries in real time to a major breaking-news event rather than the conventional publication of mathematical research.

~~== First odd perfect number ==~~

The first odd perfect number, designated <math>N</math>, and approximately equal to <math>1.1274354961045\ldots \times {10}^{1,353,382}</math>, is the first discovered odd positive integer equal to the sum of its proper positive divisors. Its decimal representation contains 1,353,383 digits and ends in …94689796301. It is strongly believed to be the smallest (and hence sequentially the first odd perfect number on the number line), although verification is ongoing.

The number was discovered on 27 January 2030 by a joint team from twelve universities in the United States, Europe and Singapore, with additional support from the {{W|Great Internet Mersenne Prime Search}} (GIMPS). Although GIMPS is primarily concerned with Mersenne primes, computational techniques developed for its prime searches were adapted to identify prime factors satisfying the known constraints on odd perfect numbers. The discovery was confirmed on the early morning of 29th of January and announced to the public shortly afterwards.

The discovery resolved the approximately 2,000-year-old question of whether odd perfect numbers exist. It was made using the method of characteristics, a number-theoretic search method that traces possible factorizations along characteristic lines subject to progressively stronger necessary conditions for odd perfect numbers. The method is unrelated to the method of characteristics used for partial differential equations.

If it is proven to be the smallest odd perfect number, it will be the 36th perfect number overall in increasing numerical order, lying between the 35th even perfect number, <math>2^{1,398,268}\left(2^{1,398,269}-1\right)</math>, and the 36th even perfect number, <math>2^{2,976,220}\left(2^{2,976,221}-1\right)</math>.

~~== Second known odd perfect number and fifth known Fermat prime ==~~

The same collaboration responsible for the discovery of <math>N</math> also maintained a parallel search for unusually large primes of restricted algebraic forms. The programme had developed partly from its collaboration with the Great Internet Mersenne Prime Search (GIMPS), whose distributed-computing infrastructure and methods for testing numbers of the form <math>2^{p}-1</math> had been adapted to other exponentially defined candidates. One branch of the project concentrated on Fermat numbers,

~~: <math>F_{n}=2^{2^{n}}+1</math>,~~

using a distributed search broadly modelled on Mersenne-prime searches but modified for the arithmetic and primality testing of numbers of the form <math>2^{m}+1</math>. GIMPS participants and researchers were reported to have contributed computational resources, software adaptations and verification work to the project, although the Fermat-number search was administered by the odd-perfect-number collaboration rather than as part of the principal GIMPS Mersenne-prime search.

~~At 16:20:22 UTC, the collaboration announced that the Fermat number with index <math>M=3477214914839</math>~~

~~was prime. The newly discovered prime,~~

~~:<math>F_{M}=2^{2^{3477214914839}}+1=3.485272\ldots \times {10}^{6.292\ldots \times {10}^{1046745990736}}</math>,~~

ending in …2625709057, was the first Fermat prime discovered since the seventeenth century and the first known example beyond the five classical Fermat primes <math>F_{0},F_{1},F_{2},F_{3}</math> and <math>F_{4}</math>. Its discovery was regarded as particularly unexpected because all previously resolved Fermat numbers above <math>F_{4}</math> had been found to be composite, beginning with Euler's factorization of <math>F_{5} = 4294967297</math> in 1732.

Owing to its enormous index, <math>F_{M}</math> was also, by an overwhelming margin, the largest known prime number at the time of its discovery. Its size exceeded not only the Mersenne primes that had historically dominated records for the largest known prime, but also the two primes G-dand G+ddiscovered earlier that day around the first elite number. Whereas those numbers contained approximately 136 billion decimal digits, the number of digits of <math>F_{M}</math> was itself of order <math>{10}^{1.046 \times 10^{12}}</math>, placing the new record on an entirely different numerical scale. Contemporary accounts consequently described the previous largest-known-prime records as effectively incomparable in magnitude.

The discovery subsequently became directly connected with the second major result of the odd-perfect-number programme. The collaboration's software automatically incorporated <math>F_{M}</math> into searches for admissible Euler factors of odd perfect numbers. Fourteen minutes after the Fermat-prime announcement, at 16:34:45 UTC, this search produced the second known odd perfect number, <math>N_{2}</math>, in which <math>F_{M}</math> occurred as the Euler special prime: <math>N_{2}=F_{M}^{65537}m^{2}</math>.

~~This made the Fermat-prime result one of the principal links in the chain of otherwise unrelated discoveries that later characterized the Day of Mathematics.~~

The discovery also had an immediate consequence in classical Euclidean geometry. By the Gauss–Wantzel theorem, a regular <math>n</math>-gon is constructible using only an unmarked straightedge and compass when <math>n</math> is the product of a power of two and distinct Fermat primes. The primality of <math>F_M</math> therefore established the existence of an entirely new (albeit an unimaginably large) family of constructible regular polygons involving <math>F_M</math> as a factor, including the regular <math>F_M</math>-gon itself. It was the first expansion of the known set of Fermat-prime factors available for constructible polygons since Gauss's 1796 discovery of the constructibility of the regular 17-gon. Although an explicit straightedge-and-compass construction of a polygon with the number of sides of <math>F_M</math> magnitude was of no practical geometric significance, the result attracted considerable attention as an unusual instance in which a computational discovery involving a number of extraordinary size immediately enlarged a classification originating in ancient Greek geometry.

The exact ordinal position of <math>N_2</math> among the odd perfect numbers (or perfect numbers overall, for that matter) remains unknown and is considered beyond current methods of determination. Preliminary work on their distribution suggested that the magnitude of the k-th odd perfect number grows at least approximately as <math>c^{c^{k}}</math>, while proposed upper estimates conjecture growth on a tetrational scale, <math>c \uparrow \uparrow k</math>, for a constant <math>c</math> whose value remains undetermined. The extreme sparsity implied by these estimates has also led to the conjecture that, if infinitely many odd perfect numbers exist, the sum of their reciprocals would be a {{W|Liouville number}}.

~~Seng subsequently applied inversion of computational science (ICS) to the reciprocal sum of all perfect numbers, without distinguishing between even and odd members,~~

~~:<math>\sum_{\begin{matrix}n\geq 1 \\ \sigma \left(n\right)=2n\end{matrix}} \frac{1}{n}=0.204520142838\ldots</math>,~~

and reported a proof that the resulting constant is {{W|transcendental}}. The argument was described as parity-agnostic: rather than depending on the known structure of even perfect numbers or the newly established properties of odd perfect numbers separately, it treated perfectness itself, <math>\sigma \left(n\right)=2n</math>, as the defining arithmetic property of the sequence. Using PSLQ- and LLL-based computational techniques within the ICS framework, Seng worked from the assumption that the reciprocal sum was algebraic of arbitrary finite degree and derived constraints on its computable structure incompatible with the arithmetic information encoded by the constant. The result was presented as an ontological application of ICS, in the sense that the argument depended on what integers qualify as perfect numbers rather than on their parity or a particular parametrization of either class. Consequently, if the proof is correct, transcendence of the full reciprocal sum also establishes that the set of perfect numbers cannot be finite, and hence that at least one of the even or odd classes must contain infinitely many members, without determining which. The proof is currently awaiting independent review.

~~== First elite number ==~~

The first elite number, <math>G=2.368259249829\ldots \times {10}^{136,248,339,111}</math>, is the first discovered counterexample to the Goldbach conjecture and therefore the first known even integer greater than 2 that cannot be represented as the sum of two odd prime numbers. Its discovery disproved the conjecture.

<math>G</math> is a colossally abundant number, and hence by definition – a superabundant number. Based on the estimated distributions of superabundant and colossally abundant numbers, <math>G</math> is predicted to be approximately the 824 billionth superabundant number and the 26 millionth colossally abundant number. Its exact ordinal positions in either sequence are unknown, as <math>G</math> was identified directly rather than through an exhaustive enumeration of all preceding superabundant and colossally abundant numbers. It also holds the record for the greatest known distance between a superabundant number and its nearest prime (typically, such numbers are only 1 away from nearest prime, like 5040 and 5039). The nearest primes lie at

~~:<math>G-31131211131221</math> and <math>G+31131211131221</math>,~~

It remains unknown whether <math>G</math> is the smallest elite number. Heuristic models instead predict that the majority of elite numbers, if there is infinitely many of them, have comparatively sparse factorizations, typically consisting of a small power of 2 multiplied by several very large distinct prime factors,

~~:<math>E=2^{a}p_{1}p_{2}\cdots p_{k}</math>,~~

where <math>a</math> is relatively small and the <math>p_{i}</math> occur predominantly to the first power. However, such candidates are, given predicted sizes of elite numbers – are impossible to identify using existing methods. Hence, the method responsible for its discovery is primarily capable of identifying elite numbers not amongst that set – but rather only among superabundant or similarly highly divisible integers exhibiting anomalously large distances from nearby primes, and consequently samples only a restricted subset of possible elite numbers. The search therefore concentrated principally on the numbers whose highly structured factorizations permitted much more efficient screening.

~~=== Search and discovery ===~~

The term elite number was coined by Mehmet Aytaç, Matthew Powell and Antonio Kʼawiil at University College Dublin. It originated in a recurring joke in their early work that anthropomorphized highly divisible composite numbers as an “aristocracy” and primes as “peasants”. Kʼawiil, a Maya mathematician from National Autonomous University of Mexico, extended the joke through a parodic form of Maya numerology, playing on popular fascination with the Maya calendar and supposed Maya mathematical mysticism: highly composite numbers were described as worthy of ceremonial use in the Long Count, while indivisible primes were “unworthy of worship”. Under the analogy, the Goldbach conjecture forced every even integer to nevertheless “associate” with prime numbers. A hypothetical counterexample was consequently termed an elite number — an even integer that did not have to “associate with the peasants”.

On 24 October 2026, Aytaç, Powell and Kʼawiil at University College Dublin, together with researchers at the National Autonomous University of Mexico and seven universities in California, began co-chairing an international computational programme devoted to the search for elite numbers. The programme marked a departure from earlier approaches to the Goldbach conjecture by inquiring into the existence of elite numbers, rather than attempting to prove the conjecture in the positive formulation, as its principal research question.

The reformulation changed the direction of the search. Rather than attempting to verify the Goldbach conjecture for progressively larger ranges of even integers, the programme sought to determine which classes of integers could contain elite numbers and to eliminate classes for which their existence could be excluded. The researchers described this approach as an inversion of the conventional problem: instead of asking whether every even integer possessed a Goldbach representation, they asked directly whether any even integer possessed none.

The computational programme eventually incorporated 6,769 supercomputers and artificial-intelligence data centres, with substantial portions of the computation devoted to the factorization, divisor structure and Goldbach representation counts of selected families of integers. Early heuristic models developed in late 2026 predicted that an elite number was most likely to possess a comparatively sparse prime factorization, consisting of a relatively small power of 2 and a small number of very large odd prime factors, with a wide separation between their logarithmic sizes. The models additionally suggested an upper bound on the exponent of 2 occurring in such a factorization. Numbers having these characteristics were, however, among the most computationally difficult candidates to investigate directly, particularly when their odd prime factors were extremely large.

The collaboration consequently adopted the initially counterintuitive strategy of examining classes in which elite numbers were considered least likely to occur. These families generally possessed highly regular factorizations and could therefore be studied substantially more efficiently. The purpose of these searches was not primarily to discover an elite number within them, but to establish increasingly broad exclusion results, progressively restricting the possible arithmetic form of any counterexample.

One of the first major investigations concerned perfect powers of even integers. Already before 2026, it was known that if <math>g(n)</math>denotes the number of Goldbach representations of an even integer n, numerical experiments had indicated that, for a fixed even integer n,

~~<math>\underset{k\rightarrow \infty }{\lim}\frac{g\left(n^{k+1}\right)}{g\left(n^{k}\right)}=n</math>.~~

~~For powers of 2, for example,~~

~~<math>\underset{k\rightarrow \infty }{\lim}\frac{g\left(2^{k+1}\right)}{g\left(2^{k}\right)}=2</math>.~~

The relationship had previously been regarded as a heuristic asymptotic property of Goldbach representation counts. In July 2027, researchers associated with the programme proved the limiting relation and strengthened it sufficiently to establish that perfect powers of even integers could not be elite numbers. Hence, this only left the integers of the form <math>2^{a}m^{b}</math>, with <math>m</math>odd and <math>a≠b</math>, and later on more general factorizations <math>2^{a}\prod_{i} p_{i}^{k_{i}}</math>. The programme nevertheless continued to investigate highly structured families considered unfavorable to elite numbers. This strategy ultimately became important to the discovery of <math>G</math>, despite the results that most elite numbers are not similar to <math>G</math> in structure.

~~== Disproof of the Riemann hypothesis ==~~

Shortly after the discovery of the first elite number, G, was announced in Berkeley, researchers studying its divisor structure determined that it was also the first known integer greater than 5,040 to violate Robin's inequality,

~~: <math>\sigma \left(n\right)\lt e^{\gamma }n\operatorname{\log{\log}}{n}</math>.~~

Because Robin's theorem establishes that the inequality holds for every <math>n>5040</math> if and only if the Riemann hypothesis is true, the result constituted an indirect disproof of the hypothesis before any non-trivial zero outside the critical line had been identified. The Berkeley team subsequently contacted Coraline Seng and Luke Bannon, whose group operated Solace, an artificial-intelligence research system developed specifically for investigations of the Riemann hypothesis.

Seng and Bannon later recalled in interviews that they initially “couldn't quite believe what they were seeing”, particularly because the result appeared to resolve the hypothesis through an unexpected divisor-theoretic consequence of <math>G</math>, rather than through the direct discovery of an exceptional zero. Following independent checks of the Robin inequality calculation, Seng and Bannon retuned Solace to search explicitly for zeros outside the critical line, incorporating Gtogether with other exceptional numbers reported earlier that day. Solace autonomously identified the first odd perfect number, <math>N</math>, as a second relevant divisor-theoretic object and tested combinations involving the two newly discovered numbers. At 16:15:34 UTC, it reported

~~:<math>\boxed{\zeta \left(\frac{1}{N}+Gi\right)=0,}</math>~~

providing the first explicit non-trivial zero known to lie outside <math>\Re (s)=\frac{1}{2}</math>. While the violation of Robin's inequality had already done so, the aforementioned complex number supplied the direct zeta-function counterexample that researchers had sought immediately afterward. The rapid progression from the discovery of <math>G</math>, through the unexpected Robin violation, to Solace's identification of an explicit zero became one of the most widely reported episodes of the Day of Mathematics.

~~== Disproof of Collatz conjecture ==~~

Later that day, Kindred, an artificial-intelligence research system operated since 2028 with the objective of resolving the Collatz conjecture, independently processed the factorization data produced following the discovery of <math>N_{2}</math>. The analysis was conducted principally by Kindred servers operated at the Massachusetts Institute of Technology (MIT). As part of its automated search, the system ingested all 10,572,384,227 prime factors identified in the factorization of the second known odd perfect number and examined their Collatz trajectories and associated parity structures.

~~At 16:40:00 UTC, Kindred identified the second prime factor of <math>N_{2}</math>,~~

~~:<math>C\approx 2.158158\times {10}^{5.1284\ldots \times {10}^{383838472427}}</math>,~~

as a counterexample to the Collatz conjecture. Rather than eventually entering the <math>4 \rightarrow 2 \rightarrow 1</math> cycle, the orbit of <math>C</math> was shown to be unbounded. Kindred subsequently produced a proof that the trajectory continued to grow without bound, establishing Cas the first known '''non-Collatzian number'''. The result was particularly unexpected because <math>C</math> had not been identified through a direct search over integers: it had entered Kindred's analysis only as one of the prime factors of <math>N_{2}</math>, discovered minutes earlier.

~~=== Second counterexample ot the Riemann hypothesis ===~~

The Kindred result was automatically incorporated into the continuing computations performed by Solace. The system combined <math>C</math> with <math>F_{M}</math>, the newly discovered Fermat prime that had independently been identified as the special prime of <math>N_{2}</math>. At 16:46:24 UTC, six minutes after Kindred's announcement, Solace reported a second non-trivial zero outside the critical line,

~~:<math>\boxed{\zeta \left(\frac{1~~}~~{F_{M~~}~~}+Ci\right)=0.}</math>~~

The result attracted particular attention because both coordinates of the new zero were derived from exceptional prime factors of the same odd perfect number: its real component was determined by <math>F_{M}</math>, while its imaginary component was the newly identified Collatz counterexample <math>C</math>. It followed the first explicit counterexample,

~~<math>\zeta \left(\frac{1}{N_{1}}+Gi\right)=0<math> reported by Solace at 16:15:34 UTC, giving an interval of only 30 minutes and 50 seconds between the discovery of the two off-critical-line zeros.~~

Bannon later stated that he was present in the office when confirmation of the second zero arrived. According to his account, Simon Decourcy, head of the Berkeley Department of Mathematics, approached him shortly after the result had been assessed through the Kindred and Solace systems and announced, ''“The Collatz is down, and a second counterexample has hit the Riemann hypothesis.”'' The remark, widely reproduced in subsequent reporting, became one of the best-known quotations associated with the Day of Mathematics. Its resemblance to breaking-news language was subsequently amplified in online coverage, where the rapid succession of automated alerts from Kindred and Solace was frequently parodied as “a second hypothesis has been hit.”

@@ Line 1: / Line 1: @@
-On 29 January 2030, a series of major mathematical breakthroughs were announced within hours of one another, resolving several of the longest-standing open problems in mathematics.
+{{SandBox please edit after this line}}
-The events began at approximately 08:00–09:00 UTC, when a multinational research collaboration spanning twelve universities in seven countries announced the discovery and verification of the first known {{W|odd perfect number}}, designated <math>N</math>, with an approximate value of <math>N=1.1274354961045…×10^{1,353,382}</math>. The discovery established for the first time that odd perfect numbers exist, resolving a problem whose origins date to antiquity.
-At 12:24 UTC, an independent collaboration involving fourteen universities announced the discovery of the first 'elite number', the term used by the project for an even integer greater than 2 that cannot be expressed as the sum of two prime numbers and therefore constitutes a counterexample to the {{W|Goldbach conjecture}}. The number, designated <math>G</math>, was approximately <math>G=2.368259249829…×10^{136,248,339,111}</math>. The discovery resulted from a reformulation of the collaboration's original research programme. Rather than attempting to establish the Goldbach conjecture universally, the project had increasingly treated its negation as a direct search problem, asking whether any elite numbers existed. The identification and subsequent verification of <math>G</math> disproved the conjecture.
-At 16:15:34 UTC, Solace, an artificial-intelligence system operated by a research team led by Luke Bannon and Coraline Seng and designed to investigate the Riemann hypothesis, autonomously incorporated <math>N</math> and <math>G</math> into its ongoing search. The system reported that
-<math>\zeta \left(\frac{1}{N}+Gi\right)=0</math>,
-together with the corresponding zeros implied by the symmetries of the Riemann zeta function. Because <math>1/N\neq \frac{1}{2}</math>, the result constituted a counterexample to the Riemann hypothesis. Independent verification of the zero subsequently established the hypothesis to be false, resolving what had widely been regarded as the foremost outstanding problem in mathematics.
-At 16:20:22 UTC, the first {{W|Fermat prime}} greater than 65,537 was announced. The number, <math>F_{M}=2^{2^{3477214914839}}+1=3.485272\ldots \times {10}^{6.292\ldots \times {10}^{1046745990736}},\</math> ends in …2625709057. The number is important as it is the number of sides of the first new constructible polygon found since 65537-gon. Contemporary media frequently referred to the number as <math>F_{5}</math>, owing to its description as the first Fermat prime discovered after the five classical examples. This notation was mathematically ambiguous, as <math>F_{5}</math> conventionally denotes the fifth-indexed Fermat ''number'', <math>2^{2^{5}}+1=4294967297</math>, which is composite. Nevertheless, extensive early publicity resulted in <math>F_{5}</math> becoming a common informal designation for the newly discovered prime. The alternative notation <math>F_{M}</math> was subsequently adopted in mathematical literature to distinguish it from the conventional Fermat-number indexing.
-At 16:34:45 UTC, software used in the ongoing search for odd perfect numbers incorporated <math>F_{M}</math> into its search parameters and identified it as the special prime of a second odd perfect number, subsequently designated <math>N_{2}</math>. In abridged notation, <math>N_{2}\approx 6.189237272\times {10}^{5.4181\times {10}^{1046746112244}}</math>, with final digits …3737117649. In the Eulerian factorization of <math>N_{2}</math>, <math>F_{M}</math> occurs to the 65,537th power, <math>N_{2}=F_{M}^{65537}m^{2}</math>, where <math>m</math> is odd and coprime to <math>F_{M}</math>. The exponent 65,537 is itself a Fermat prime, coincidentally.
-In parallel, at 16:40:00 UTC, Kindred, an artificial-intelligence research system operated since 2028 with the objective of resolving the {{W|Collatz conjecture}}, incorporated the newly reported factorization of <math>N_{2}</math> into its search. Kindred determined that the second prime factor of <math>N_{2}</math>, subsequently designated <math>C</math>, was a counterexample to the conjecture. Its Collatz orbit was shown to be unbounded and therefore never to reach 1. The result constituted the first known non-Collatzian positive integer and disproved the Collatz conjecture.
-At 16:46:24 UTC, the values of <math>F_{M}</math> and <math>C</math> were incorporated into a further search conducted by Solace. The system subsequently identified a second non-trivial zero of the Riemann zeta function lying outside the critical line, satisfying <math>\zeta \left(\frac{1}{F_{M}}+Ci\right)=0</math>. The discovery provided a second explicit counterexample to the Riemann hypothesis, following the zero identified earlier that day. The real and imaginary components of the new zero were thus determined respectively by two prime factors of <math>N_{2}</math>, linking the Fermat-prime, odd-perfect-number, Collatz and Riemann-hypothesis results within the same sequence of discoveries.
-The extraordinary concentration of major results on 29 January 2030 was widely regarded as unprecedented in the history of mathematics. Within a single day, several long-standing problems—including the existence of odd perfect numbers, Goldbach's conjecture, the Collatz conjecture and the Riemann hypothesis—were resolved or overturned, alongside the discovery of the first Fermat prime since the seventeenth century. The event was considered particularly remarkable because the results were not merely simultaneous: successive discoveries repeatedly supplied the mathematical objects used in later ones, with the newly identified numbers <math>N_{1}</math>, <math>G</math>, <math>F_{M}</math>, <math>N_{2}</math> and <math>C</math> becoming interconnected across otherwise largely unrelated fields of number theory. The resulting sequence of announcements received extensive international media coverage and became known collectively as the Day of Mathematics, with contemporary commentary frequently comparing the experience of following the discoveries in real time to a major breaking-news event rather than the conventional publication of mathematical research.
-== First odd perfect number ==
-The first odd perfect number, designated <math>N</math>, and approximately equal to <math>1.1274354961045\ldots \times {10}^{1,353,382}</math>, is the first discovered odd positive integer equal to the sum of its proper positive divisors. Its decimal representation contains 1,353,383 digits and ends in …94689796301. It is strongly believed to be the smallest (and hence sequentially the first odd perfect number on the number line), although verification is ongoing.
-The number was discovered on 27 January 2030 by a joint team from twelve universities in the United States, Europe and Singapore, with additional support from the {{W|Great Internet Mersenne Prime Search}} (GIMPS). Although GIMPS is primarily concerned with Mersenne primes, computational techniques developed for its prime searches were adapted to identify prime factors satisfying the known constraints on odd perfect numbers. The discovery was confirmed on the early morning of 29th of January and announced to the public shortly afterwards.
-The discovery resolved the approximately 2,000-year-old question of whether odd perfect numbers exist. It was made using the method of characteristics, a number-theoretic search method that traces possible factorizations along characteristic lines subject to progressively stronger necessary conditions for odd perfect numbers. The method is unrelated to the method of characteristics used for partial differential equations.
-If it is proven to be the smallest odd perfect number, it will be the 36th perfect number overall in increasing numerical order, lying between the 35th even perfect number, <math>2^{1,398,268}\left(2^{1,398,269}-1\right)</math>, and the 36th even perfect number, <math>2^{2,976,220}\left(2^{2,976,221}-1\right)</math>.
-== Second known odd perfect number and fifth known Fermat prime ==
-The same collaboration responsible for the discovery of <math>N</math> also maintained a parallel search for unusually large primes of restricted algebraic forms. The programme had developed partly from its collaboration with the Great Internet Mersenne Prime Search (GIMPS), whose distributed-computing infrastructure and methods for testing numbers of the form <math>2^{p}-1</math> had been adapted to other exponentially defined candidates. One branch of the project concentrated on Fermat numbers,
-: <math>F_{n}=2^{2^{n}}+1</math>,
-using a distributed search broadly modelled on Mersenne-prime searches but modified for the arithmetic and primality testing of numbers of the form <math>2^{m}+1</math>. GIMPS participants and researchers were reported to have contributed computational resources, software adaptations and verification work to the project, although the Fermat-number search was administered by the odd-perfect-number collaboration rather than as part of the principal GIMPS Mersenne-prime search.
-At 16:20:22 UTC, the collaboration announced that the Fermat number with index <math>M=3477214914839</math>
-was prime. The newly discovered prime,
-:<math>F_{M}=2^{2^{3477214914839}}+1=3.485272\ldots \times {10}^{6.292\ldots \times {10}^{1046745990736}}</math>,
-ending in …2625709057, was the first Fermat prime discovered since the seventeenth century and the first known example beyond the five classical Fermat primes <math>F_{0},F_{1},F_{2},F_{3}</math> and <math>F_{4}</math>. Its discovery was regarded as particularly unexpected because all previously resolved Fermat numbers above <math>F_{4}</math> had been found to be composite, beginning with Euler's factorization of <math>F_{5} = 4294967297</math> in 1732.
-Owing to its enormous index, <math>F_{M}</math> was also, by an overwhelming margin, the largest known prime number at the time of its discovery. Its size exceeded not only the Mersenne primes that had historically dominated records for the largest known prime, but also the two primes G-dand G+ddiscovered earlier that day around the first elite number. Whereas those numbers contained approximately 136 billion decimal digits, the number of digits of <math>F_{M}</math> was itself of order <math>{10}^{1.046 \times 10^{12}}</math>, placing the new record on an entirely different numerical scale. Contemporary accounts consequently described the previous largest-known-prime records as effectively incomparable in magnitude.
-The discovery subsequently became directly connected with the second major result of the odd-perfect-number programme. The collaboration's software automatically incorporated <math>F_{M}</math> into searches for admissible Euler factors of odd perfect numbers. Fourteen minutes after the Fermat-prime announcement, at 16:34:45 UTC, this search produced the second known odd perfect number, <math>N_{2}</math>, in which <math>F_{M}</math> occurred as the Euler special prime: <math>N_{2}=F_{M}^{65537}m^{2}</math>.
-This made the Fermat-prime result one of the principal links in the chain of otherwise unrelated discoveries that later characterized the Day of Mathematics.
-The discovery also had an immediate consequence in classical Euclidean geometry. By the Gauss–Wantzel theorem, a regular <math>n</math>-gon is constructible using only an unmarked straightedge and compass when <math>n</math> is the product of a power of two and distinct Fermat primes. The primality of <math>F_M</math> therefore established the existence of an entirely new (albeit an unimaginably large) family of constructible regular polygons involving <math>F_M</math> as a factor, including the regular <math>F_M</math>-gon itself. It was the first expansion of the known set of Fermat-prime factors available for constructible polygons since Gauss's 1796 discovery of the constructibility of the regular 17-gon. Although an explicit straightedge-and-compass construction of a polygon with the number of sides of <math>F_M</math> magnitude was of no practical geometric significance, the result attracted considerable attention as an unusual instance in which a computational discovery involving a number of extraordinary size immediately enlarged a classification originating in ancient Greek geometry.
-The exact ordinal position of <math>N_2</math> among the odd perfect numbers (or perfect numbers overall, for that matter) remains unknown and is considered beyond current methods of determination. Preliminary work on their distribution suggested that the magnitude of the k-th odd perfect number grows at least approximately as <math>c^{c^{k}}</math>, while proposed upper estimates conjecture growth on a tetrational scale, <math>c \uparrow \uparrow k</math>, for a constant <math>c</math> whose value remains undetermined. The extreme sparsity implied by these estimates has also led to the conjecture that, if infinitely many odd perfect numbers exist, the sum of their reciprocals would be a {{W|Liouville number}}.
-Seng subsequently applied inversion of computational science (ICS) to the reciprocal sum of all perfect numbers, without distinguishing between even and odd members,
-:<math>\sum_{\begin{matrix}n\geq 1 \\ \sigma \left(n\right)=2n\end{matrix}} \frac{1}{n}=0.204520142838\ldots</math>,
-and reported a proof that the resulting constant is {{W|transcendental}}. The argument was described as parity-agnostic: rather than depending on the known structure of even perfect numbers or the newly established properties of odd perfect numbers separately, it treated perfectness itself, <math>\sigma \left(n\right)=2n</math>, as the defining arithmetic property of the sequence. Using PSLQ- and LLL-based computational techniques within the ICS framework, Seng worked from the assumption that the reciprocal sum was algebraic of arbitrary finite degree and derived constraints on its computable structure incompatible with the arithmetic information encoded by the constant. The result was presented as an ontological application of ICS, in the sense that the argument depended on what integers qualify as perfect numbers rather than on their parity or a particular parametrization of either class. Consequently, if the proof is correct, transcendence of the full reciprocal sum also establishes that the set of perfect numbers cannot be finite, and hence that at least one of the even or odd classes must contain infinitely many members, without determining which. The proof is currently awaiting independent review.
-== First elite number ==
-The first elite number, <math>G=2.368259249829\ldots \times {10}^{136,248,339,111}</math>, is the first discovered counterexample to the Goldbach conjecture and therefore the first known even integer greater than 2 that cannot be represented as the sum of two odd prime numbers. Its discovery disproved the conjecture.
-<math>G</math> is a colossally abundant number, and hence by definition – a superabundant number. Based on the estimated distributions of superabundant and colossally abundant numbers, <math>G</math> is predicted to be approximately the 824 billionth superabundant number and the 26 millionth colossally abundant number. Its exact ordinal positions in either sequence are unknown, as <math>G</math> was identified directly rather than through an exhaustive enumeration of all preceding superabundant and colossally abundant numbers. It also holds the record for the greatest known distance between a superabundant number and its nearest prime (typically, such numbers are only 1 away from nearest prime, like 5040 and 5039). The nearest primes lie at
-:<math>G-31131211131221</math> and <math>G+31131211131221</math>,
-It remains unknown whether <math>G</math> is the smallest elite number. Heuristic models instead predict that the majority of elite numbers, if there is infinitely many of them, have comparatively sparse factorizations, typically consisting of a small power of 2 multiplied by several very large distinct prime factors,
-:<math>E=2^{a}p_{1}p_{2}\cdots p_{k}</math>,
-where <math>a</math> is relatively small and the <math>p_{i}</math> occur predominantly to the first power. However, such candidates are, given predicted sizes of elite numbers – are impossible to identify using existing methods. Hence, the method responsible for its discovery is primarily capable of identifying elite numbers not amongst that set – but rather only among superabundant or similarly highly divisible integers exhibiting anomalously large distances from nearby primes, and consequently samples only a restricted subset of possible elite numbers. The search therefore concentrated principally on the numbers whose highly structured factorizations permitted much more efficient screening.
-=== Search and discovery ===
-The term elite number was coined by Mehmet Aytaç, Matthew Powell and Antonio Kʼawiil at University College Dublin. It originated in a recurring joke in their early work that anthropomorphized highly divisible composite numbers as an “aristocracy” and primes as “peasants”. Kʼawiil, a Maya mathematician from National Autonomous University of Mexico, extended the joke through a parodic form of Maya numerology, playing on popular fascination with the Maya calendar and supposed Maya mathematical mysticism: highly composite numbers were described as worthy of ceremonial use in the Long Count, while indivisible primes were “unworthy of worship”. Under the analogy, the Goldbach conjecture forced every even integer to nevertheless “associate” with prime numbers. A hypothetical counterexample was consequently termed an elite number — an even integer that did not have to “associate with the peasants”.
-On 24 October 2026, Aytaç, Powell and Kʼawiil at University College Dublin, together with researchers at the National Autonomous University of Mexico and seven universities in California, began co-chairing an international computational programme devoted to the search for elite numbers. The programme marked a departure from earlier approaches to the Goldbach conjecture by inquiring into the existence of elite numbers, rather than attempting to prove the conjecture in the positive formulation, as its principal research question.
-The reformulation changed the direction of the search. Rather than attempting to verify the Goldbach conjecture for progressively larger ranges of even integers, the programme sought to determine which classes of integers could contain elite numbers and to eliminate classes for which their existence could be excluded. The researchers described this approach as an inversion of the conventional problem: instead of asking whether every even integer possessed a Goldbach representation, they asked directly whether any even integer possessed none.
-The computational programme eventually incorporated 6,769 supercomputers and artificial-intelligence data centres, with substantial portions of the computation devoted to the factorization, divisor structure and Goldbach representation counts of selected families of integers. Early heuristic models developed in late 2026 predicted that an elite number was most likely to possess a comparatively sparse prime factorization, consisting of a relatively small power of 2 and a small number of very large odd prime factors, with a wide separation between their logarithmic sizes. The models additionally suggested an upper bound on the exponent of 2 occurring in such a factorization. Numbers having these characteristics were, however, among the most computationally difficult candidates to investigate directly, particularly when their odd prime factors were extremely large.
-The collaboration consequently adopted the initially counterintuitive strategy of examining classes in which elite numbers were considered least likely to occur. These families generally possessed highly regular factorizations and could therefore be studied substantially more efficiently. The purpose of these searches was not primarily to discover an elite number within them, but to establish increasingly broad exclusion results, progressively restricting the possible arithmetic form of any counterexample.
-One of the first major investigations concerned perfect powers of even integers. Already before 2026, it was known that if <math>g(n)</math>denotes the number of Goldbach representations of an even integer n, numerical experiments had indicated that, for a fixed even integer n,
-<math>\underset{k\rightarrow \infty }{\lim}\frac{g\left(n^{k+1}\right)}{g\left(n^{k}\right)}=n</math>.
-For powers of 2, for example,
-<math>\underset{k\rightarrow \infty }{\lim}\frac{g\left(2^{k+1}\right)}{g\left(2^{k}\right)}=2</math>.
-The relationship had previously been regarded as a heuristic asymptotic property of Goldbach representation counts. In July 2027, researchers associated with the programme proved the limiting relation and strengthened it sufficiently to establish that perfect powers of even integers could not be elite numbers. Hence, this only left the integers of the form <math>2^{a}m^{b}</math>, with <math>m</math>odd and <math>a≠b</math>, and later on more general factorizations <math>2^{a}\prod_{i} p_{i}^{k_{i}}</math>. The programme nevertheless continued to investigate highly structured families considered unfavorable to elite numbers. This strategy ultimately became important to the discovery of <math>G</math>, despite the results that most elite numbers are not similar to <math>G</math> in structure.
-== Disproof of the Riemann hypothesis ==
-Shortly after the discovery of the first elite number, G, was announced in Berkeley, researchers studying its divisor structure determined that it was also the first known integer greater than 5,040 to violate Robin's inequality,
-: <math>\sigma \left(n\right)\lt e^{\gamma }n\operatorname{\log{\log}}{n}</math>.
-Because Robin's theorem establishes that the inequality holds for every <math>n>5040</math> if and only if the Riemann hypothesis is true, the result constituted an indirect disproof of the hypothesis before any non-trivial zero outside the critical line had been identified. The Berkeley team subsequently contacted Coraline Seng and Luke Bannon, whose group operated Solace, an artificial-intelligence research system developed specifically for investigations of the Riemann hypothesis.
-Seng and Bannon later recalled in interviews that they initially “couldn't quite believe what they were seeing”, particularly because the result appeared to resolve the hypothesis through an unexpected divisor-theoretic consequence of <math>G</math>, rather than through the direct discovery of an exceptional zero. Following independent checks of the Robin inequality calculation, Seng and Bannon retuned Solace to search explicitly for zeros outside the critical line, incorporating Gtogether with other exceptional numbers reported earlier that day. Solace autonomously identified the first odd perfect number, <math>N</math>, as a second relevant divisor-theoretic object and tested combinations involving the two newly discovered numbers. At 16:15:34 UTC, it reported
-:<math>\boxed{\zeta \left(\frac{1}{N}+Gi\right)=0,}</math>
-providing the first explicit non-trivial zero known to lie outside <math>\Re (s)=\frac{1}{2}</math>. While the violation of Robin's inequality had already done so, the aforementioned complex number supplied the direct zeta-function counterexample that researchers had sought immediately afterward. The rapid progression from the discovery of <math>G</math>, through the unexpected Robin violation, to Solace's identification of an explicit zero became one of the most widely reported episodes of the Day of Mathematics.
-== Disproof of Collatz conjecture ==
-Later that day, Kindred, an artificial-intelligence research system operated since 2028 with the objective of resolving the Collatz conjecture, independently processed the factorization data produced following the discovery of <math>N_{2}</math>. The analysis was conducted principally by Kindred servers operated at the Massachusetts Institute of Technology (MIT). As part of its automated search, the system ingested all 10,572,384,227 prime factors identified in the factorization of the second known odd perfect number and examined their Collatz trajectories and associated parity structures.
-At 16:40:00 UTC, Kindred identified the second prime factor of <math>N_{2}</math>,
-:<math>C\approx 2.158158\times {10}^{5.1284\ldots \times {10}^{383838472427}}</math>,
-as a counterexample to the Collatz conjecture. Rather than eventually entering the <math>4 \rightarrow 2 \rightarrow 1</math> cycle, the orbit of <math>C</math> was shown to be unbounded. Kindred subsequently produced a proof that the trajectory continued to grow without bound, establishing Cas the first known '''non-Collatzian number'''. The result was particularly unexpected because <math>C</math> had not been identified through a direct search over integers: it had entered Kindred's analysis only as one of the prime factors of <math>N_{2}</math>, discovered minutes earlier.
-=== Second counterexample ot the Riemann hypothesis ===
-The Kindred result was automatically incorporated into the continuing computations performed by Solace. The system combined <math>C</math> with <math>F_{M}</math>, the newly discovered Fermat prime that had independently been identified as the special prime of <math>N_{2}</math>. At 16:46:24 UTC, six minutes after Kindred's announcement, Solace reported a second non-trivial zero outside the critical line,
-:<math>\boxed{\zeta \left(\frac{1}{F_{M}}+Ci\right)=0.}</math>
-The result attracted particular attention because both coordinates of the new zero were derived from exceptional prime factors of the same odd perfect number: its real component was determined by <math>F_{M}</math>, while its imaginary component was the newly identified Collatz counterexample <math>C</math>. It followed the first explicit counterexample,
-<math>\zeta \left(\frac{1}{N_{1}}+Gi\right)=0<math> reported by Solace at 16:15:34 UTC, giving an interval of only 30 minutes and 50 seconds between the discovery of the two off-critical-line zeros.
-Bannon later stated that he was present in the office when confirmation of the second zero arrived. According to his account, Simon Decourcy, head of the Berkeley Department of Mathematics, approached him shortly after the result had been assessed through the Kindred and Solace systems and announced, ''“The Collatz is down, and a second counterexample has hit the Riemann hypothesis.”'' The remark, widely reproduced in subsequent reporting, became one of the best-known quotations associated with the Day of Mathematics. Its resemblance to breaking-news language was subsequently amplified in online coverage, where the rapid succession of automated alerts from Kindred and Solace was frequently parodied as “a second hypothesis has been hit.”