Some Conjectures Related to Fermat's Last Theorem

Granville, Andrew; Mollin, R. A.

doi:10.1515/9783110848632-017

Cited by 30 publications

(20 citation statements)

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“…For example, H. W. Lenstra [15] conjectures that in fact p ≤ 3; this has been supported by extensive computation, see [4,13]. The motivation to the conjecture (4) comes from the fact that this has some interesting applications to the Fermat Last Theorem [7,Corollary 1]. Although this motivation relating p to the Fermat Last Theorem does not exist anymore, improving the bounds (1) and (2) is still of interest and may have some other applications.…”

Section: (Log P)mentioning

confidence: 99%

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On the divisibility of Fermat quotients

Bourgain¹,

Ford²,

Konyagin³

et al. 2010

Michigan Math. J.

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Section: (Log P)mentioning

confidence: 99%

“…It is well known that divisibility of Fermat quotients q p (a) by p has numerous applications which include the Fermat Last Theorem and squarefreeness testing, see [5,6,7,15]. In particular, the smallest value p of a for which q p (a) ≡ 0 (mod p) plays a prominent role in these applications.…”

Section: Introductionmentioning

confidence: 99%

On the divisibility of Fermat quotients

Bourgain¹,

Ford²,

Konyagin³

et al. 2010

Michigan Math. J.

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“…There are several results which involve the distribution and structure of Fermat quotients q p (u) modulo p and it has numerous applications in computational and algebraic number theory, see e.g. [8,10,11,21] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Structure of Pseudorandom Numbers Derived from Fermat Quotients

Chen

Ostafe

Winterhof

2010

Arithmetic of Finite Fields

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Abstract.We study the distribution of s-dimensional points of Fermat quotients modulo p with arbitrary lags. If no lags coincide modulo p the same technique as in [21] works. However, there are some interesting twists in the other case. We prove a discrepancy bound which is unconditional for s = 2 and needs restrictions on the lags for s > 2. We apply this bound to derive results on the pseudorandomness of the binary threshold sequence derived from Fermat quotients in terms of bounds on the well-distribution measure and the correlation measure of order 2, both introduced by Mauduit and Sárközy. We also prove a lower bound on its linear complexity profile. The proofs are based on bounds on exponential sums and earlier relations between discrepancy and both measures above shown by Mauduit, Niederreiter and Sárközy. Moreover, we analyze the lattice structure of Fermat quotients modulo p with arbitrary lags.

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“…Many later results are surveyed in Ribenboim's book [24]. Granville [10,11], besides proving new results, provides a review of known facts and open problems.…”

Section: Introductionmentioning

confidence: 99%

“…Note that d p is necessarily prime. Granville [10,11] proved, as a slight improvement upon a result by Lenstra [19], that d p < log 2 p. The main argument in both authors' works is again the simple fact that the total number of zeros in the matrix M p is known; in addition, some analytical results about the distribution of prime numbers are required.…”

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confidence: 99%