## Wieferich’s criterion and the abc-conjecture.(English)Zbl 0654.10019

The following result is proved. The so called “abc-conjecture” of Masser and Oesterlé implies that the number of primes less than $$X$$ for which $$\alpha^{p-1}\not\equiv 1 \pmod{p^2}$$ where $$\alpha$$ is a fixed rational number and $$\alpha \neq \pm 1,0$$, is at least $$O(\log X)$$. An analogous result is also proved for points of infinite order on elliptic curves having certain $$j$$-invariants. The proofs base on several skillful lemmas.

### MathOverflow Questions:

Examples of theorems where numerical bounds on $$\pi$$ played a role

### MSC:

 11D41 Higher degree equations; Fermat’s equation 14H52 Elliptic curves
Full Text:

### Online Encyclopedia of Integer Sequences:

Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1.

### References:

 [1] Apostol, T., (Introduction to Analytic Number Theory (1976), Springer-Verlag: Springer-Verlag New York) · Zbl 0335.10001 [2] Granville, A., Powerful numbers and Fermat’s last theorem, C.R. Math. Rep. Acad. Sci. Canada, 8, 215-218 (1986) · Zbl 0595.10011 [3] Granville, A.; Monagan, M., 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, 329-359 (1988) · Zbl 0645.10018 [4] Johnson, W., On the non-vanishing of the Fermat quotient modulo $$p$$, J. Reine Angew. Math., 292, 196-200 (1977) · Zbl 0347.10001 [5] Brillhart, J.; Tonascia, J.; Weinberger, P., On the Fermat quotient, (Atkin, A. O.L.; Birch, B. J., Computers in Number Theory (1971), Academic Press: Academic Press New York/London), 213-222 · Zbl 0217.03203 [6] Ribenboim, R., (13 Lectures on Fermat’s Last Theorem (1979), Springer-Verlag: Springer-Verlag New York) · Zbl 0456.10006 [7] Silverman, J. H., (The Arithmetic of Elliptic Curves (1986), Springer-Verlag: Springer-Verlag New York) · Zbl 0585.14026 [8] Silverman, J. H., Integral points on curves and surfaces, (Journées Arithmétiques—Ulm (1987), Springer-Verlag), in press · Zbl 0979.11037 [9] Wieferich, A., Zum letzten Fermat’schen Theorem, J. Reine Angew. Math., 136, 293-302 (1909)
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