Computation of an Improved Lower Bound to Giuga's Primality Conjecture
نویسنده
چکیده
Our most recent computations tell us that any counterexample to Giuga’s 1950 primality conjecture must have at least 19,908 decimal digits. Equivalently, any number which is both a Giuga and a Carmichael number must have at least 19,908 decimal digits. This bound has not been achieved through exhaustive testing of all numbers with up to 19,908 decimal digits, but rather through exploitation of the properties of Giuga and Carmichael numbers. This bound improves upon the 1996 bound of one of the authors. We present the algorithm used, and our improved bound. We also discuss the changes over the intervening years as well as the challenges to further computation.
منابع مشابه
Giuga's Conjecture on Primality
G. Giuga conjectured that if an integer n satisses n?1 P k=1 k n?1 ?1 mod n, then n must be a prime. We survey what is known about this interesting and now fairly old conjecture. Giuga proved that n is a counterexample to his conjecture if and only if each prime divisor p of n satisses (p ? 1) j (n=p ? 1) and p j (n=p ? 1). Using this characterization, he proved computationally that any counter...
متن کاملGiuga's Conjecture on Primality
k=l Fermat's little theorem says that if p is a prime, then kP-l 3 lmodp for k = 1,.. .,p-1. Therefore, for each prime p, Sp--lmodp. The question becomes: Does there xist a non-prime n such that Sn 3 1 mod n? This question has resisted solution for more than forty years. After surveying what is known about the conjecture, we will give several new results here which might suggest directions of f...
متن کاملA bound for Feichtinger conjecture
In this paper, using the discrete Fourier transform in the finite-dimensional Hilbert space C^n, a class of nonRieszable equal norm tight frames is introduced and using this class, a bound for Fiechtinger Conjecture is presented. By the Fiechtinger Conjecture that has been proved recently, for given A,C>0 there exists a universal constant delta>0 independent of $n$ such that every C-equal...
متن کاملThe lower bound for the number of 1-factors in generalized Petersen graphs
In this paper, we investigate the number of 1-factors of a generalized Petersen graph $P(N,k)$ and get a lower bound for the number of 1-factors of $P(N,k)$ as $k$ is odd, which shows that the number of 1-factors of $P(N,k)$ is exponential in this case and confirms a conjecture due to Lovász and Plummer (Ann. New York Acad. Sci. 576(2006), no. 1, 389-398).
متن کاملOn a conjecture of a bound for the exponent of the Schur multiplier of a finite $p$-group
Let $G$ be a $p$-group of nilpotency class $k$ with finite exponent $exp(G)$ and let $m=lfloorlog_pk floor$. We show that $exp(M^{(c)}(G))$ divides $exp(G)p^{m(k-1)}$, for all $cgeq1$, where $M^{(c)}(G)$ denotes the c-nilpotent multiplier of $G$. This implies that $exp( M(G))$ divides $exp(G)$, for all finite $p$-groups of class at most $p-1$. Moreover, we show that our result is an improvement...
متن کامل