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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.
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| 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.
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| 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
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| <math>\zeta \left(\frac{1}{N}+Gi\right)=0</math>,
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| 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.
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| At 16:20:22 UTC, the first 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. 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.
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