Amicable pairs and aliquot cycles on average
نویسندگان
چکیده
منابع مشابه
Amicable Pairs and Aliquot Cycles for Elliptic Curves
An amicable pair for an elliptic curve E/Q is a pair of primes (p, q) of good reduction for E satisfying #Ẽp(Fp) = q and #Ẽq(Fq) = p. In this paper we study elliptic amicable pairs and analogously defined longer elliptic aliquot cycles. We show that there exist elliptic curves with arbitrarily long aliqout cycles, but that CM elliptic curves (with j 6= 0) have no aliqout cycles of length greate...
متن کاملAmicable Pairs and Aliquot Sequences
This constant can, in fact, be rigorously calculated to 149 digits (and probably much higher accuracy if needed). Define () to be the th iterate of with starting value . The integer is amicable or 2-sociable if () = but () 6= . Such phrasing is based on older terminology [3]: two distinct integers , are said to form an “amicable pair” if () = and () = . The (infinite?) ...
متن کاملAmicable Pairs and Aliquot Cycles for Elliptic Curves over Number Fields
Let E/Q be an elliptic curve. Silverman and Stange define primes p and q to be an elliptic amicable pair if #E(Fp) = q and #E(Fq) = p. More generally, they define the notion of aliquot cycles for elliptic curves. Here we study the same notion in the case that the elliptic curve is defined over a number field K. We focus on proving the existence of an elliptic curve E/K with aliquot cycle (p1, ....
متن کاملOn Φ–amicable Pairs
Let φ(n) denote Euler’s totient function, i.e., the number of positive integers < n and prime to n. We study pairs of positive integers (a0, a1) with a0 ≤ a1 such that φ(a0) = φ(a1) = (a0 + a1)/k for some integer k ≥ 1. We call these numbers φ–amicable pairs with multiplier k, analogously to Carmichael’s multiply amicable pairs for the σ–function (which sums all the divisors of n). We have comp...
متن کاملAliquot Cycles of Repdigits
Here we show that the only aliquot cycle consisting only of rep-digits in base 10 is the cycle consisting of the perfect number 6. Generally, we show that if g is an even positive integer, then there are only finitely many aliquot cycles consisting entirely of repdigits in base g, which are, at least in principle, effectively computable.
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ژورنال
عنوان ژورنال: International Journal of Number Theory
سال: 2015
ISSN: 1793-0421,1793-7310
DOI: 10.1142/s1793042115500761