A343716 Numbers k such that k^2 divides 5^k - 4^k - 3^k.

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

Offset: 1

Author

Jon E. Schoenfield, May 08 2021

Keywords

• nonn
• more

Comments

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.

a(41) > 10^11 if it exists. - Chai Wah Wu, May 16 2021

Examples

			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.
		

Crossrefs

Cf. A015616, A127074, A343115.

Programs

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=", ")
afind(limit=10**5) # Michael S. Branicky, May 16 2021