A172426 - OEIS

%I #21 Jul 30 2023 02:43:33

%S 5,12,27,75,363,1587,2523,5043,8427,20667,23763,38307,51483,89787,

%T 96123,109443,162867,171363,189003,236883,257547,386643,526683,557283,

%U 588747,723243,777243,1054947,1232643,1279227,1302843,1399467,1550883,1656147,1737363,1963443,2489763,2724627

%N Number of nontrivial solutions (x,y,z) for each prime number p of the Fermat equation x^p + y^p + z^p = 0 mod (n) where n is prime of the form n = 2p + 1, and x, y, z are integers such that x < = y.

%C Solution to a Diophantine equation in finite fields Z/n. Historical reminder: Sophie Germain's work led to the breaking of Fermat's Last Theorem into two cases: x^p + y^p = z^p has no integer solutions for which x, y and z are relatively prime to p, i.e., in which none of x, y and z are divisible by p, and then x^p + y^p = z^p has no integer solutions for which one of the three numbers is divisible by p.

%C This result was presented by Legendre in an 1823 paper to the French Academy of Sciences and included in a supplement to his second edition of Theorie des Nombres, with a footnote crediting the result to Sophie Germain. Sophie Germain's Theorem introduced an auxiliary prime n satisfying the two conditions: x^p + y^p + z^p == 0 (mod n) implies that x == 0 (mod n), or y == 0 (mod n), or z == 0 (mod n), and x^p == p (mod n) is impossible for any value of x. Then Case I of Fermat's Last Theorem is true for p. This sequence gives solutions for each prime number p, and n = 2p + 1.

%D René Schoof, "Wiles' proof of the Taniyama-Weil conjecture for semi-stable elliptic curves over Q", Chap. 14 in 'Où en sont les Mathématiques ?' Soc. Math. de France (SMF), Vuibert, Paris 2002.

%H Robin Visser, <a href="/A172426/b172426.txt">Table of n, a(n) for n = 1..1000</a>

%H C. K. Caldwell, The Prime Glossary,<a href="https://t5k.org/glossary/page.php?sort=FermatsLastTheorem"> Fermat's Last Theorem</a>

%H Andrea Del Centina, <a href="http://web.unife.it/progetti/geometria/Germain.html">Letters of Sophie Germain preserved in Florence</a>, Historia Mathematica, Vol. 32 (2005), 60-75.

%H Andrea Del Centina, <a href="https://www.jstor.org/stable/41134283">Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem</a>, Arch. Hist. Exact Sci., Vol 62 (2008), 349-392.

%H A. M. Legendre, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k3221x/f189.item">Recherches sur quelques objets d'analyse indéterminée et particulièrement sur le théorème de Fermat</a>, Mem. Acad. Sci. Inst. France 6 (1823), 1-60.

%H J. H. Sampson, <a href="https://www.jstor.org/stable/41133883">Sophie Germain and the theory of numbers</a>, Arch. Hist. Exact Sci. 41 (1990), 157-161.

%e We consider the case p = 1, n = 3. We have 5 solutions mod 3: (0,1,2), (0,2,1), (1,1,1), (1,2,0), (2,2,2).

%e With p = 2, n = 5, we have 12 solutions mod 5: (0,1,2), (0,1,3), (0,2,1), (0,2,4), (0,3,1), (0,3,4), (0,4,2), (0,4,3), (1,2,0), (1,3,0), (2,4,0), (3,4,0),

%e With p = 3, n = 7, we have 27 solutions mod 7: (0,1,3), (0,1,5), (0,1,6), (0,2,3), (0,2,5), (0,2,6), (0,3,1), (0,3,2), (0,3,4), (0,4,3), (0,4,5), (0,4,6), (0,5,1), (0,5,2), (0,5,4), (0,6,1), (0,6,2), (0,6,4), (1,3,0), (1,5,0), (1,6,0), (2,3,0), (2,5,0), (2,6,0), (3,4,0), (4,5,0), (4,6,0).

%o (Sage)

%o p = 1

%o while (p < 1000):

%o     n, ans = 2*p + 1, 0

%o     numz = [0 for i in range(n)]

%o     for i in range(n): numz[power_mod(i,p,n)] += 1

%o     for y in range(1,n):

%o         for x in range(y+1):

%o             ans += numz[(-power_mod(x,p,n)-power_mod(y,p,n))%n]

%o     print(ans)

%o     p = p.next_prime()

%o     while(not (2*p+1).is_prime()): p = p.next_prime()  # _Robin Visser_, Jul 29 2023

%Y Cf. A019590 (Fermat's last theorem).

%K nonn

%O 1,1

%A _Michel Lagneau_, Feb 02 2010

%E More terms from _Robin Visser_, Jul 29 2023