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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.
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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.

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