User:Eliora/Sandbox: Difference between revisions

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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.

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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.

@@ Line 11: / Line 11: @@
 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.
@@ Line 24: / Line 26: @@
 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.