On congruences related to the first case of Fermat’s last theorem

Author:
Wells Johnson

Journal:
Math. Comp. **31** (1977), 519-526

MSC:
Primary 10B15; Secondary 10A10

DOI:
https://doi.org/10.1090/S0025-5718-1977-0447108-8

MathSciNet review:
0447108

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Solutions to the congruences ${(1 + a)^{{p^n}}} \equiv 1 + {a^{{p^n}}}\pmod {p^{n + 2}}$ and ${(1 + s)^p} \equiv 1 + {s^p}\pmod {p^n}$ are discussed. Congruences of this type arise in the study of the first case of Fermat’s Last Theorem. Solutions to these congruences always exist for primes $p \equiv 1\;\pmod 6$. They are derived from the existence of a primitive cube root of unity $\pmod p$. Constructive techniques for finding numerical examples are presented. The results are obtained by examining the *p*-adic expansions of the *p*-adic $(p - 1)$st roots of unity.

- A. Arwin,
*Über die Lösung der Kongruenz $(\lambda +I)p-l\lambda p-I-o (mod p^2)$*, Acta Math.**42**(1920), no. 1, 173–190 (German). MR**1555163**, DOI https://doi.org/10.1007/BF02404406 - R. D. Carmichael,
*Note on Fermat’s last theorem*, Bull. Amer. Math. Soc.**19**(1913), no. 5, 233–236. MR**1559329**, DOI https://doi.org/10.1090/S0002-9904-1913-02332-2
L. E. DICKSON, - Ioanna Ferentinou-Nikolakopoulou,
*A new necessary condition for the existence of a solution to the equation $x^{p}+y^{p}=z^{p}$ of Fermat*, Bull. Soc. Math. Grèce (N.S.)**6 I**(1965), no. fasc. 2, 222–236 (Greek, with French summary). MR**205912**
J. M. GANDHI, "Fermat’s Last Theorem I. Some interesting observations for the first case," - Wells Johnson,
*$p$-adic proofs of congruences for the Bernoulli numbers*, J. Number Theory**7**(1975), 251–265. MR**376512**, DOI https://doi.org/10.1016/0022-314X%2875%2990020-7
W. MEISSNER, "Über die Lösungen der Kongruenz ${x^{p - 1}} \equiv 1\;\bmod {p^m}$ und ihre Verwertung zur Periodenbestimmung $\bmod {p^x}$," - H. S. Vandiver,
*Note on Fermat’s last theorem*, Trans. Amer. Math. Soc.**15**(1914), no. 2, 202–204. MR**1500973**, DOI https://doi.org/10.1090/S0002-9947-1914-1500973-0

*History of the Theory of Numbers*, vol. II, Carnegie Institution of Washington, Washington, D.C., 1920. C. J. EVERETT & N. METROPOLIS, "On the roots of ${x^m} \pm 1$ in the

*p*-adic field ${Q_p}$,"

*Notices Amer. Math. Soc.*, v. 22, 1975, p. A-619. Abstract #75T-A229.

*Notices Amer. Math. Soc.*, v. 22, 1975, p. A-486. Abstract #725-A2. J. M. GANDHI, "On the first case of Fermat’s Last Theorem." (To appear.)

*Sitzungsber. Berlin Math. Gessell.*, v. 13, 1914, pp. 96-107. F. POLLACZEK, "Über den grossen Fermat’schen Satz,"

*Sitzungsber. Akad. Wiss. Wien*(

*Math.*), v. 126 (IIa), 1917, pp. 45-59. A. A. TRYPANIS, "On Fermat’s last theorem,"

*Proc. Internat. Congr. of Mathematicians*(Cambridge, Mass., 1950), vol. 1, Amer. Math. Soc., Providence, R.I., 1952, pp. 301-302.

Retrieve articles in *Mathematics of Computation*
with MSC:
10B15,
10A10

Retrieve articles in all journals with MSC: 10B15, 10A10

Additional Information

Keywords:
Congruences,
Fermat’s Last Theorem,
<I>p</I>-adic roots of unity

Article copyright:
© Copyright 1977
American Mathematical Society