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## Acta Arithmetica

1991-1992 | 60 | 4 | 371-388
Tytuł artykułu

### Fibonacci numbers and Fermat's last theorem

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let {Fₙ} be the Fibonacci sequence defined by F₀=0, F₁=1, $F_{n+1}=Fₙ+F_{n-1} (n≥1)$. It is well known that $F_{p-(5/p)}≡ 0 (mod p)$ for any odd prime p, where (-) denotes the Legendre symbol. In 1960 D. D. Wall [13] asked whether $p²|F_{p-(5/p)}$ is always impossible; up to now this is still open.
In this paper the sum $∑_{k≡ r (mod 10)}{n\choose k}$ is expressed in terms of Fibonacci numbers. As applications we obtain a new formula for the Fibonacci quotient $F_{p-(5/p)}/p$ and a criterion for the relation $p|F_{(p-1)/4}$ (if p ≡ 1 (mod 4), where p ≠ 5 is an odd prime. We also prove that the affirmative answer to Wall's question implies the first case of FLT (Fermat's last theorem); from this it follows that the first case of FLT holds for those exponents which are (odd) Fibonacci primes or Lucas primes.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
371-388
Opis fizyczny
Daty
wydano
1992
otrzymano
1990-11-27
poprawiono
1991-05-16
Twórcy
autor
• Department of Mathematics Nanjing University Nanjing 210008 People's Republic of China
Bibliografia
• [1] L. E. Dickson, History of the Theory of Numbers, Vol. I, Chelsea, New York 1952, 105, 393-396.
• [2] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, Oxford 1981, 148-150.
• [3] E. Lehmer, On the quartic character of quadratic units, J. Reine Angew. Math. 268/269 (1974), 294-301.
• [4] L. J. Mordell, Diophantine Equations, Academic Press, London and New York 1969, 60-61.
• [5] P. Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, New York 1979, 139-159.
• [6] Zhi-Hong Sun, Combinatorial sum $∑^n_{k=0 k≡ r (mod m)}{n \choose k}$ and its applications in number theory (I), J. Nanjing Univ. Biquarterly, in press.
• [7] Zhi-Hong Sun, Combinatorial sum $∑^n_{k=0 k≡ r (mod m)}{n \choose k}$ and its applications in number theory (II), J. Nanjing Univ. Biquarterly, in press.
• [8] Zhi-Wei Sun, A congruence for primes, preprint, 1991.
• [9] Zhi-Wei Sun, On the combinatorial sum $∑_{k≡ r (mod m)}{n \choose k}$, submitted.
• [10] Zhi-Wei Sun, Combinatorial sum $∑_{k≡ r (mod 12)}{n \choose k}$ and its number-theoretical applications, to appear.
• [11] Zhi-Wei Sun, Reduction of unknowns in Diophantine representations, Science in China (Ser. A) 35 (1992), 1-13.
• [12] H. S. Vandiver, Extension of the criteria of Wieferich and Mirimanoff in connection with Fermat's last theorem, J. Reine Angew. Math. 144 (1914), 314-318.
• [13] D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly 67 (1960), 525-532.
• [14] H. C. Williams, A note on the Fibonacci quotient $F_{p-ε}/p$ , Canad. Math. Bull. 25 (1982), 366-370
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Bibliografia
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