Fibonacci numbers with the Lehmer property

• Autorzy: Luca, F.
• Języki publikacji: EN

Abstrakty

EN

We show that if m > 1 is a Fibonacci number such that φ(m) | m - 1, where φ is the Euler function, then m is prime.

Słowa kluczowe

EN

Fibonacci number; Lucas number; Euler function; Lehmer's conjecture

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2007
• Tom: Vol. 55, no 1
• Strony: 7-15
• Opis fizyczny: Bibliogr. 18 poz.

Twórcy

autor

Luca, F.

• Institute de Matematicas Universidad Nacional Autonoma de Mexico, Ap. Postal 61-3 (Xangari), C.P. 58089, Morelia, Michoacan, Mexico,

Bibliografia

• [1] V. Andreji, On Fibonacci powers, Univ. Beograd Publ. Elektrotehn. Fak. Sec. Mat. 17 (2006), 38-44.
• [2] W. D. Banks and F. Luca, Composite integers n for which φ(n) | n — 1, Acta Math. Sinica, to appear.
• [3] Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. (2) 163 (2006), 969-1018.
• [4] R. D. Carmichael, On the numerical factors of the arithmetic forms αn±βn, ibid. 15 (1913), 30-70.
• [5] G. L. Cohen and P. Hagis, On the number of prime factors of n if φ(n) | n — 1, Nieuw Arch. Wisk. (3) 28 (1980), 177-185.
• [6] M. Diaconescu, On the equation m — 1 = aφ(m), Integers 6 (2006), art. 6.
• [7] R. K. Guy, Unsolved Problems in Number Theory, Springer, 2004.
• [8] M. Kishore, On the equation kφ(M) = M - 1, Nieuw Arch. Wisk. (3) 25 (1977), 48-53.
• [9] D. H. Lehmer, On Euler's totient function, Bull. Amer. Math. Soc. 38 (1932), 745-751.
• [10] E. Lieuwens, Do there exist composite numbers for which kφ(M) = M — 1 holds? Nieuw Arch. Wisk. (3) 18 (1970), 165-169.
• [11] F. Luca and L. Szalay, Fibonacci numbers of the form pa ±pb +1, Fibonacci Quart., to appear.
• [12] R. J. Mclntosh and E. L. Roettger, A search for Fibonacci-Wiefrich and Wolsen-holme primes, preprint, 2006.
• [13] H. L. Montgomery and R. C. Vaughan, The large sieve,- Mathematika 20 (1973), 119-134.
• [14] C. Pomerance, On composite n for which φ(n) | n-1, Acta Arith. 28 (1976), 387-389.
• [15] —, On composite n for which φ(n) | n-1, II, Pacific J. Math. 69 (1977), 177-186.
• [16] Z. H. Sun and Z. W. Sun, Fibonacci numbers and Fermat's last theorem, Acta Arith. 60 (1992), 371-388.
• [17] D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly 67 (1960), 525-532.
• [18] S. Zun, On composite n for which φ(n) | n - 1, J. China Univ. Sci. Tech. 15 (1985), 109-112.