OFFSET
1,1
COMMENTS
Related to Legendre's contribution to Fermat's last theorem: the first case of Fermat's last theorem is true only if the Diophantine equation x^n + y^n = z^n has integer solutions x,y,z where n is prime such that gcd(n, xyz) = 1, then n >= 197.
REFERENCES
A. E. Bojarincev, Asymptotic expressions for the n-th composite number, Univ. Mat. Zap. 6:21-43 (1967). - In Russian.
J. M. De Koninck, Ces nombres qui nous fascinent, Ellipses, 2008, p. 59.
Sampson, J.H. "Sophie Germain and the theory of numbers," Arch. Hist. Exact Sci. 41 (1990), 157-161.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fermat's Last Theorem
EXAMPLE
With p=197 we obtain the composite numbers 2p+1 = 5*79, 4p+1 = 3*263, 8p+1 = 19*83, 10p+1 = 27*73, 14p+1 = 31*89 and 16p+1 = 3*1051.
MAPLE
for p from 1 to 2000 do: if type(p, prime)=true and type(2*p+1, prime)=false and type(4*p+1, prime)=false and type(8*p+1, prime)=false and type(10*p+1, prime)=false and type(14*p+1, prime)=false and type(16*p+1, prime)=false then print(p):else fi:od:
MATHEMATICA
aQ[p_] := PrimeQ[p] && AllTrue[{2 p + 1, 4 p + 1, 8 p + 1, 10 p + 1, 14 p + 1, 16 p + 1}, CompositeQ]; Select[Range[1400], aQ] (* Amiram Eldar, Aug 10 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Apr 04 2010
STATUS
approved