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%I #8 Mar 31 2012 10:23:48
%S 1147,113,113,400,866,131,399,32,26,29,23,58,77,21,42,3,817,4,2,37,80,
%T 29,181,39,120,382,76,5,29,20,48,19,36,7,43,7,62,22,7,43,5,17,23,44,
%U 52,137,103,2,5,49,31,10,30,5,25,25,49,10,72,50,13,4,7,6
%N a(n) is the smallest number m such that no prime takes the form of n^m+/-n^k+/-1, while 0 <= k < m and m > 1.
%C Conjecture: a(n) has finite value when a>4
%C already tested: a(4)>2364; a(3)>7399; and a(2)>9594.
%C Hypothesis is that a(2), a(3), and a(4) are infinite.
%C Mathematica program ran about an hour and gave the first 96 items.
%C When n is larger, a(n) tends to be 2 for most of n.
%e n=5, there is no prime number in the form of 5^1147+/-5^k+/-1 for 0 <= k < 1147
%t Table[i = 1; While[i++; c1 = b^i; cs = {};
%t Do[c2 = b^j; cp = c1 + c2 + 1;
%t If[PrimeQ[cp], cs = Union[cs, {cp}]];
%t cp = c1 + c2 - 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
%t cp = c1 - c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
%t cp = c1 - c2 - 1;
%t If[PrimeQ[cp], cs = Union[cs, {cp}]], {j, 0, i - 1}];
%t ct = Length[cs]; ct > 0]; i, {b, 5, 100}]
%Y Cf. A196697, A196698, A196778.
%K nonn
%O 5,1
%A _Lei Zhou_, Oct 06 2011
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