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A343716
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Numbers k such that k^2 divides 5^k - 4^k - 3^k.
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0
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1, 2, 4, 6, 8, 10, 12, 18, 24, 26, 30, 36, 50, 72, 74, 90, 130, 150, 338, 370, 450, 650, 962, 1402, 1850, 2738, 2974, 4810, 8450, 12506, 24050, 35594, 64382, 68450, 184706, 270150, 312650, 462722, 889850, 11568050
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OFFSET
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1,2
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COMMENTS
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If it exists, a(41) > 10^9.
Of the 4*A015616(10) = 4*109 = 436 integer sequences of one of the forms
Numbers k such that k^2 | A^k + B^k + C^k,
Numbers k such that k^2 | A^k + B^k - C^k,
Numbers k such that k^2 | A^k - B^k + C^k,
or Numbers k such that k^2 | A^k - B^k - C^k
such that 0 < C < B < A <= 10 and gcd(A,B,C)=1, this one appears to have the largest number of terms.
By comparison, A127074 (k such that k^2 | 3^k - 2^k - 1) and A343115 (k such that k^2 | 5^k - 3^k - 2^k) seem unlikely to have any terms beyond A127074(9)=17807 and A343115(14)=876, respectively. Only 25 of the 436 above sequences have any 4-, 5-, or 6-digit terms at all.
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LINKS
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EXAMPLE
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5^2 - 4^2 - 3^2 = 25 - 16 - 9 = 0, which is divisible by 2^2 = 4, so 2 is a term.
5^18 - 4^18 - 3^18 = 3745590368400 = 11560464100 * 18^2, so 18 is a term.
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PROG
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(Python)
def afind(startat=1, limit=10**9):
for k in range(startat, limit+1):
kk = k*k
if (pow(5, k, kk) - pow(4, k, kk) - pow(3, k, kk))%kk == 0:
print(k, end=", ")
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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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STATUS
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approved
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