A survey of results relating to Giuga’s
                    conjecture on primality

Borwein, Jonathan M.; Wong, Erick B.

doi:10.1090/crmp/011/02

Cited by 5 publications

(4 citation statements)

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“…We observe that the set Q 10 = {3, 5,7,11,13,17,19,23, 29} is the smallest set of odd primes whose reciprocal sum is greater than 1. Thus a counterexample must, because |Q 10 | = 9, have at least 9 prime factors and, because p∈Q10 p > 10 9 , must have at least 10 digits.…”

Section: Notation 1 (Odd Primes and Sets Of Odd Primes)mentioning

confidence: 96%

Computation of an Improved Lower Bound to Giuga’s Primality Conjecture

Skerritt

2014

Mathematical Software – ICMS 2014

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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.

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Section: Notation 1 (Odd Primes and Sets Of Odd Primes)mentioning

confidence: 96%

Computation of an Improved Lower Bound to Giuga’s Primality Conjecture

Skerritt

2014

Mathematical Software – ICMS 2014

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“…Much work has been done regarding these numbers [1,3,10,13,16,17] and several generalizations and/or variations are possible [6,9]. In this paper some new characterizations of Giuga numbers will be established.…”

Section: José María Grau a M Oller-marcén 2015mentioning

confidence: 99%

Variations on Giuga Numbers and Giuga’s Congruence

Grau

Oller-Marcén

2016

Ukr Math J

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A k-strong Giuga number is a composite integer such that n−1 j=1 j n−1 ≡ −1 (mod n). We consider the congruence n−1 j=1 j k(n−1) ≡ −1 (mod n) for each k ∈ N (thus extending Giuga's ideas for k = 1). In particular, it is proved that a pair (n, k) with composite n satisfies this congruence if and only if n is a Giuga number and λ(n) | k(n − 1). In passing, we establish some new characterizations of Giuga numbers and study some properties of the numbers n satisfying λ(n) | k(n − 1). k-Сильне число Гюга-це складене цiле число таке, що n−1 j=1 j n−1 ≡ −1 (mod n). Ми розглядаємо конгруенцiю n−1 j=1 j k(n−1) ≡ −1 (mod n) для кожного k ∈ N (таким чином ми розширюємо iдеї Гюга на випадок k = 1). Як частинний випадок доведено, що пара (n, k) зi складеним n задовольняє цю конгруенцiю тодi i тiльки тодi, коли n-число Гюга та λ(n) | k(n − 1). Крiм того, встановлено деякi новi характеристики чисел Гюга та вивчено властивостi чисел n, що задовольняють λ(n) | k(n − 1).

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“…Other unsolved problems about Giuga sequences are listed in Borwein et al [2,Sect. 4] and in Borwein and Wong [1].…”

Section: An Allowable Value Is Mmentioning

confidence: 99%

Cyclic Systems of Simultaneous Congruences

Lagarias

2010

Int. J. Number Theory

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This paper considers the cyclic system of n ≥ 2 simultaneous congruencesfor fixed nonzero integers (r, s) with r > 0 and (r, s) = 1. It shows there are only finitely many solutions in positive integers q i ≥ 2, with gcd(q 1 q 2 · · · qn, s) = 1 and obtains sharp bounds on the maximal size of solutions for almost all (r, s). The extremal solutions for r = s = 1 are related to Sylvester's sequence 2, 3, 7, 43, 1807, . . . . If the positivity condition on the integers q i is dropped, then for r = 1 these systems of congruences, taken (mod |q i |), have infinitely many solutions, while for r ≥ 2 they have finitely many solutions. The problem is reduced to studying integer solutions of the family of Diophantine equationsdepending on three parameters (r, s, m).

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A survey of results relating to Giuga’s conjecture on primality

Abstract: This article is an expanded version of the talk given by the rst author at the 25th Anniversary Conference of the Centre de R echerches Math ematiques.

Cited by 5 publications

References 0 publications

Computation of an Improved Lower Bound to Giuga’s Primality Conjecture

Computation of an Improved Lower Bound to Giuga’s Primality Conjecture

Variations on Giuga Numbers and Giuga’s Congruence

Cyclic Systems of Simultaneous Congruences

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