Fibonacci numbers and Fermat's last theorem

Acta Arithmetica (1992)

• Volume: 60, Issue: 4, page 371-388
• ISSN: 0065-1036

top

Abstract

top
Let Fₙ be the Fibonacci sequence defined by F₀=0, F₁=1, ${F}_{n+1}=Fₙ+{F}_{n-1}\left(n\ge 1\right)$. It is well known that ${F}_{p-\left(5/p\right)}\equiv 0\left(modp\right)$ for any odd prime p, where (-) denotes the Legendre symbol. In 1960 D. D. Wall [13] asked whether $p²|{F}_{p-\left(5/p\right)}$ is always impossible; up to now this is still open. In this paper the sum ${\sum }_{k\equiv r\left(mod10\right)}\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ is expressed in terms of Fibonacci numbers. As applications we obtain a new formula for the Fibonacci quotient ${F}_{p-\left(5/p\right)}/p$ and a criterion for the relation $p|{F}_{\left(p-1\right)/4}$ (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.

How to cite

top

Zhi-Wei Sun. "Fibonacci numbers and Fermat's last theorem." Acta Arithmetica 60.4 (1992): 371-388. <http://eudml.org/doc/206445>.

@article{Zhi1992,
abstract = {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 $∑_\{k≡ r (mod 10)\}\{n\atopwithdelims ()k\}$ 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.},
author = {Zhi-Wei Sun},
journal = {Acta Arithmetica},
keywords = {Fibonacci numbers; Legendre symbol; Lucas numbers; Fibonacci quotient; Fermat's last theorem; Fibonacci primes; Lucas primes},
language = {eng},
number = {4},
pages = {371-388},
title = {Fibonacci numbers and Fermat's last theorem},
url = {http://eudml.org/doc/206445},
volume = {60},
year = {1992},
}

TY - JOUR
AU - Zhi-Wei Sun
TI - Fibonacci numbers and Fermat's last theorem
JO - Acta Arithmetica
PY - 1992
VL - 60
IS - 4
SP - 371
EP - 388
AB - 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 $∑_{k≡ r (mod 10)}{n\atopwithdelims ()k}$ 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.
LA - eng
KW - Fibonacci numbers; Legendre symbol; Lucas numbers; Fibonacci quotient; Fermat's last theorem; Fibonacci primes; Lucas primes
UR - http://eudml.org/doc/206445
ER -

References

top
1. [1] L. E. Dickson, History of the Theory of Numbers, Vol. I, Chelsea, New York 1952, 105, 393-396.
2. [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. [3] E. Lehmer, On the quartic character of quadratic units, J. Reine Angew. Math. 268/269 (1974), 294-301. Zbl0289.12007
4. [4] L. J. Mordell, Diophantine Equations, Academic Press, London and New York 1969, 60-61.
5. [5] P. Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, New York 1979, 139-159.
6. [6] Zhi-Hong Sun, Combinatorial sum ${\sum }_{k=0k\equiv r\left(modm\right)}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ and its applications in number theory (I), J. Nanjing Univ. Biquarterly, in press.
7. [7] Zhi-Hong Sun, Combinatorial sum ${\sum }_{k=0k\equiv r\left(modm\right)}^{n}\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ and its applications in number theory (II), J. Nanjing Univ. Biquarterly, in press.
8. [8] Zhi-Wei Sun, A congruence for primes, preprint, 1991.
9. [9] Zhi-Wei Sun, On the combinatorial sum ${\sum }_{k\equiv r\left(modm\right)}\left(\genfrac{}{}{0pt}{}{n}{k}\right)$, submitted.
10. [10] Zhi-Wei Sun, Combinatorial sum ${\sum }_{k\equiv r\left(mod12\right)}\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ and its number-theoretical applications, to appear.
11. [11] Zhi-Wei Sun, Reduction of unknowns in Diophantine representations, Science in China (Ser. A) 35 (1992), 1-13.
12. [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. Zbl45.0289.02
13. [13] D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly 67 (1960), 525-532. Zbl0101.03201
14. [14] H. C. Williams, A note on the Fibonacci quotient ${F}_{p-\epsilon }/p$ , Canad. Math. Bull. 25 (1982), 366-370 Zbl0491.10009

top

NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.