Wieferich Primes and Hall’s Conjecture

Satya Mohit & M. Ram Murty

Abstract Nous montrons que la Conjecture de Hall pour courbes elliptiques implique l’existence d’une infinité de nombres premiers p satisfaisant a^{p−1} ≢ 1 (mod p^{16}), pour chaque nombre entier a.

In the long and sinuous history of Fermat’s last theorem, the following criterion stands out:

If a prime p satisfies

2^{p−1} ≢ 1 (mod p^{2}),

then the first case of Fermat’s last theorem is true for the exponent p.

This criterion, proven by Wieferich [1] in 1909, was subsequently generalized by a number of authors to allow 2 to be replaced by any prime q ≤ 89. (See [2].) Computational evidence shows that the only primes p ≤ 4 × 10^{12} satisfying 2^{p−1} ≡ 1 (mod p^{2}) are p = 1093 and 3511. Furthermore, for any integer a, the primes satisfying a^{p−1} ≡ 1 (mod p^{2}) are very few. (Tables listing all solutions of a^{p−1} ≡ 1 (mod p^{2}) for 2 ≤ a ≤ 99, 3 ≤ p ≤ 2^{32} are listed in [3], p.347-349. The smallest known example where this congruence is satisfied is the case of 3^{10} ≡ 1 (mod 11^{2}).) However, the following conjecture remains outstanding: Conjecture 1. For any a ∈ ℤ, there exist infinitely many primes p such that

a^{p−1} ≢ 1 (mod p^{2}).

It is not known whether the assertion holds for any fixed integer a. Neither is the truth of the following weaker assertion. Conjecture 2. For a given a ∈ ℤ, there exists a natural number n such that there exist infinitely many primes p such that

a^{p−1} ≢ 1 (mod p^{n}).

Silverman [4] has shown that the ABC-conjecture of Masser and Oesterlé implies Conjecture 1. More precisely, he showed that the ABC-conjecture implies that

#{p ≤ x|a^{p−1} ≢ 1 (mod p^{2})} ≫_{a} log x as x → ∞.

Another consequence of the ABC-conjecture is the following conjecture of Hall [5]: Conjecture 3. (Hall): Let ϵ > 0 be given. Then for any nonzero integer D, every solution x, y ∈ ℤ to the equation

y^{2} = x^{3} + D

satisfies

|x| ≪_{ϵ} |D|^{2+ϵ}.

We will show in this note that Conjecture 3 (which is much weaker than the powerful ABC-conjecture) implies Conjecture 2. In fact, we will prove the following: Theorem. If Hall’s Conjecture (Conjecture 3) is true, then

#\{p ≤ x|a^{p−1} ≢ 1 (mod p^{16})\} ≫_{a} \frac{log x}{log log x}\}.

We require two elementary lemmas. Definition. Let n ∈ ℤ. We will call n d^{th}-powerful if, for every p|n, we have p^{d}|n. Lemma 1. Let n ∈ ℤ be d^{th}-powerful. Then n may be written in the form

n = \{ \begin{matrix}α^{2}β^{d} & if d is odd \\ α^{2}β^{d+1} & if d is even \end{matrix} ,

for some integers α, β, with β squarefree. Proof. Let n = ∏_{i∈S} p_{i}^{e_{i}} be the canonical factorization of n. By hypothesis, each e_{i} ≥ d.

If d is odd, we may write

n = ∏_{e_{i} even} p_{i}^{e_{i}} ∏_{e_{j} odd} p_{j}^{e_{j}} = ∏_{e_{i} even} p_{i}^{e_{i}} ∏_{e_{j} odd} p_{j}^{e_{j}−d} ∏_{e_{j} odd} p_{j}^{d}

= ( ∏_{e_{i} even} pi^{\frac{e_{i}}{2}} ∏_{e_{j} odd} pj^{\frac{e_{j} −d}{2}} )^{2}( ∏_{e_{j} odd} p_{j})^{d}

If d is even, write

n = ∏_{e_{i}=d} p_{i}^{e_{i}} · m,

where m = ∏_{e_{i}>d} p_{i}^{e_{i}} is (d + 1)^{th}-powerful. By the above, m = α^{2}β^{d+1}, for some α, β ∈ ℤ, β squarefree. Therefore,

n = ( ∏_{e_{i}=d} p_{i}^{d/2}α)^{2}β^{d+1}.

Lemma 2. Suppose that p^{t}∥a^{n} − 1 with t ≥ 1. (i.e., p^{t} is the highest power of p dividing a^{n} − 1.) Then a^{p−1} ≢ 1 (mod p^{t+1}). Proof. Let d denote the multiplicative order of a mod p. Then a^{n} ≡ 1 (mod p) ⇒ d|n ⇒ a^{d} − 1|a^{n} − 1 so that a^{d} ≢ 1 (mod p^{t+1}).

Let p^{s}∥a^{d} − 1 (so that 1 ≤ s ≤ t) and write a^{d} = 1 + kp^{s}, where p |̸k. Now, d|p − 1, say p − 1 = df. Thus,

a^{p−1} = (a^{d})^{f} = (1 + kp^{s})^{f}
= ∑_{i=0}^{f} \binom{f}{i}(kp^{s})^{i}
≡ 1 + fkp^{s} (mod p^{s+1})
≢ 1 (mod p^{t+1})

Proof (of Theorem). We show that, for any a ∈ ℤ, a^{3k+1} − 1 is not 16^{th}-powerful, for all k sufficiently large. By the previous lemma, this will establish that, for all p|a^{3k+1} − 1, a^{p−1} ≢ 1 (mod p^{16}).

Suppose that a^{3k+1} − 1 = α_{k}^{2}β_{k}^{d}, where d is odd. (By Lemma 1, any integer that is at least (d − 1)^{th}-powerful may be written in this form.) Then, multiplying through by a^{2}β_{k}^{3}, we obtain

a^{3k+3}β_{k}^{3} − a^{2}β_{k}^{3} = a^{2}α_{k}^{2}β_{k}^{d+3}

or

(a^{k+1}β_{k})^{3} − a^{2}β_{k}^{3} = (aα_{k}βk^{\frac{d+3}{2}} )^{2}.

Applying Hall’s Conjecture, we have

a^{k+1}β_{k} ≪_{ϵ} (a^{2}β_{k}^{3})^{2+ϵ} or a^{k+1} ≪_{ϵ,a} β_{k}^{5+ϵ}.

Then

β_{k}^{d} < α_{k}^{2}β_{k}^{d} < a^{3k+1} − 1 < (a^{k+1})^{3} ≪_{ϵ,a} β_{k}^{15+3ϵ}.

Now, for fixed β, the curve β^{d}y^{2} = ax^{3} − 1 has but finitely many integral points. Therefore, β_{k} → ∞ as k → ∞. We can thus conclude from the above inequality that d ≤ 15, for all sufficiently large k. Thus, for all sufficiently large k, a^{3k+1} − 1 is not 16^{th}-powerful⇒ for all p|a^{3k+1}−1, a^{p−1} ≢ 1 (mod p^{16}). Since for gcd(3k+1, 3l+1) = 1, gcd(a^{3k+1} − 1, a^{3l+1} − 1) = a − 1, every sufficiently large prime q ≡ 1 (mod 3) yields a distinct prime p|\frac{a^{q}−1}{a−1}. There are thus, infinitely many primes p such that a^{p−1} ≢ 1 (mod p^{16}). In fact,

#{p ≤ x|a^{p−1} ≢ 1 (mod p^{16})} ≫ #{p ≡ 1 (mod 3)|a^{p} − 1 ≤ x}
≫ #\{p ≡ 1 (mod 3)|p ≤ \frac{log x}{log a}\}
≫_{a} \frac{log x}{log log x} as x → ∞,

by the prime number theorem for arithmetic progressions.

References

1. A. Wieferich. Zum letzen Fermat’schen Theorem, J. Reine Angew. Math. 136 (1909), 293-302.

2. A. Granville and M. Monagan, The first case of Fermat’s last theorem is true for all prime exponents up to 714, 591, 416, 091, 389, Trans. Amer. Math. Soc. 306 (1988), 329-359.

3. P. Ribenboim, ”The New Book of Prime Number Records”, Springer-Verlag, New York, 1989.

4. J. H. Silverman, Wieferich’s criterion and the abc-conjecture, Journal of Number Theory 30 (1988), 226-337.

5. M. Hall Jr., The diophantine equation x^{3} − y^{2} = k, in ”Computers in Number Theory” (A. O. L. Atkin and B. J. Birch, Eds.), pp. 173-205, Academic Press, New York/London, 1971.