There are No Multiply-Perfect Fibonacci Numbers

Abstract

We show that no Fibonacci number (larger than 1) divides the sum of its divisors.

Multiperfect numbers with identical digits

“…But we have seen already that q divides V 2 i n for at most one value of i, and so the claim is proved. Now (22) and (23) yield that ω (V 2 i n ) 2 for all but O g (1) indices i with 0 i < s, as desired. 2…”

“…For every two indices i, j ∈ C, the greatest common divisor of U n i /U n i+1 and U n j /U n j+1 is supported on the primes dividing g − 1. So to prove (19), it is enough to show that for all but O g (1) …”

“…Indeed, if p is an odd prime dividing V 2 i n for some 0 i < s, then g 2 i n ≡ −1 (mod p), which implies that g n has order 2 i+1 modulo p. Consequently, the index i is uniquely determined. So it suffices to show that for all but O g (1) …”

“…The method was that of the first author [6], who had earlier shown that there are no perfect Fibonacci numbers (see also [7]). Quite recently it was shown [1] that there are no multiply perfect Fibonacci numbers. Inspired by this achievement, we establish the following results:…”

Odd Repdigits to Small Bases Are Not Perfect

“…Work in showing infinite classes of natural numbers which are not perfect or multiperfect has been developed by Luca, who has shown that no Fibonacci number is perfect [6], that no element of a Lucas sequence with odd parameters is multiperfect [7] and, with Broughan, González, Lewis, Huguet and Togbé, that no Fibonacci number is multiperfect [2].…”