A341115 Numbers k such that k*2^101 + 1 is a prime factor of 10^(10^100) + 1.

125000, 61298400, 578869250, 4511718750, 195312500000, 2918554687500, 3874552343750

Offset: 1

Author

Yan Sheng Ang, Feb 05 2021

Keywords

• nonn
• fini
• hard
• more

Comments

Every prime factor of 10^(10^100) + 1 is of the given form (k == 1 (mod 2^101)).

If k is not divisible by 10, then k == 1,3,4 (mod 10), and k*2^101 + 1 divides 10^(2^100) + 1.

If 1 <= j <= 99 and k is not divisible by 5^(j+1), then k*2^101 + 1 divides 10^(2^100*5^j) + 1.

No other terms below 4*10^12. Other known terms in this sequence are 397299146187500, 194585800170898437500, 3163315773010253906250, 3274180926382541656494140625000, 128238752949982881546020507812500, 13940493204245285596698522567749023437500, 61902333925445418572053313255310058593750, 146251500493521646717454132158309221267700195312500.

Examples

			The smallest prime factor of 10^10^100 + 1 is 125000*2^100 + 1 = 316912650057057350374175801344000001.
		

Links

• D. A. Alpern, Factors of 1000 numbers starting from googolplex

Crossrefs

Cf. A341116 (corresponding primes), A072288.

Programs

Python

A341115_list, k, m, l, n = [], 1, 2**101, 2**101+1, 10**100
while k < 10**6:
    if pow(10,n,l) == l-1:
        A341115_list.append(k)
        print(len(A341115_list),k)
    k += 1
    l += m # Chai Wah Wu, Mar 28 2021