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A268511
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Odd integers n such that 3^n + 5^n = x^2 + y^2 (x and y integers) is solvable.
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0
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1, 5, 13, 17, 29, 89, 109, 149, 157, 193, 373
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OFFSET
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1,2
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COMMENTS
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Corresponding 3^n + 5^n values are 8, 3368, 1222297448, 763068593288, 186264583553473068008, ...
445 <= a(12) <= 509. 509, 661, 709 are terms. - Chai Wah Wu, Jul 22 2020
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LINKS
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EXAMPLE
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1 is a term because 3^1 + 5^1 = 8 = 2^2 + 2^2.
5 is a term because 3^5 + 5^5 = 3368 = 2^2 + 58^2.
13 is a term because 3^13 + 5^13 = 1222297448 = 4118^2 + 34718^2.
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MATHEMATICA
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Select[Range[1, 110, 2], Resolve@ Exists[{x, y}, Reduce[3^# + 5^# == (x^2 + y^2), {x, y}, Integers]] &] (* Michael De Vlieger, Feb 07 2016 *)
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PROG
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(PARI) is(n) = #bnfisintnorm(bnfinit(z^2+1), n);
for(n=1, 1e3, if(n%2==1 && is(3^n + 5^n), print1(n, ", ")));
(Python)
from sympy import factorint
for n in range(1, 50, 2):
m = factorint(3**n+5**n)
for d in m:
if d % 4 == 3 and m[d] % 2:
break
else:
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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