The Agoh-Giuga conjecture, in number theory, concerns the Bernoulli numbers and postulates that a positive integer p is prime if and only if p times the (p-1)th Bernoulli number is congruent to negative one modulo p. It is named after Takashi Agoh and Giuseppe Giuga. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Statementp is a prime number if and only if pB_{p-1} is congruent to -1 (mod p), where B_{p-1} is a Bernoulli number. 1 Proposed YearThe original 1950 conjecture by Giuga; the 1995 paper by Agoh gave the equivalent Bernoulli-number reformulation the atlas states as the statement above. Progress Toward ResolutionAny composite counterexample to the Agoh-Giuga congruence would have to be both a Carmichael number and a Giuga number, and would need at least 13,800 digits; a 2001 search found none below 10^36067. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Agoh-Giuga Conjecture (Wikipedia)
Sources
1. Agoh-Giuga Conjecture (Wikipedia)
Wikimedia FoundationLead section
It is named after Takashi Agoh and Giuseppe Giuga.
Introduction (lead), congruence statement
p is a prime number if and only if pB_{p-1} ≡ -1 (mod p).
Equivalent formulation section
The conjecture as stated above is due to Takashi Agoh; an equivalent formulation is due to Giuseppe Giuga, from 1950,
Status section, digit-bound sentence
It has been shown that a composite number n satisfies the formula if and only if it is both a Carmichael number and a Giuga number, and that if such a number exists, it has at least 13,800 digits.
Status section, 2001 Sorini bound
In 2001, Sorini showed that a possible counterexample should be greater than 10^36067
- In Branch: Number Theory, Lead sentence
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