Amicable Numbers Research Articles

Iterated stable numbers and iterated amicable pairs

This note presents the restricted divisor functions and describes the amicable pairs and their iterations and the iterated stable numbers derived from these new functions. These arithmetic functions are related to (2)‐type Mersenne primes 2p − 1, (2, 3)‐type Mersenne primes 2i · 3j − 1 and multiperfect numbers. Using Mathematica 11.2, we have tabulated all iterated stable numbers and iterated amicable pairs up to 220. We establish several interesting arithmetical properties of the iterated restricted divisor functions. The tables obtained suggested many open problems on iterated stable numbers, and iterated amicable pairs from iteration of the restricted divisor functions, some of which are proven here. For any natural number n, we studied the order of its iteration relative to the restricted divisor functions. Moreover, the generating series of the iterated restricted divisor functions are found.

Компьютер как новая реальность математики: III. Числа Мерсенна и суммы делителей

Nowhere in mathematics is the progress resulting from the advent of computers is as apparent, as in the additive number theory. In this part, we describe the role of computers in the investigation of the oldest function studied in mathematics, the divisor sum. The disciples of Pythagoras started to systematically explore its behaviour more that 2500 years ago. A description of the trajectories of this function — perfect numbers, amicable numbers, sociable numbers, and the like — constitute the contents of several problems stated over 2500 years ago, which still seem completely inaccessible. A theorem due to Euclid and Euler reduces classification of even perfect numbers to Mersenne primes. After 1914 not a single new Mersenne prime was ever produced manually, since 1952 all of them have been discovered by computers. Using computers, now we construct hundreds or thousands times more new amicable pairs daily, than what was constructed by humans over several millenia. At the end of the paper, we discuss yet another problem posed by Catalan and Dickson

The reciprocal sum of the amicable numbers

The reciprocal sum of the amicable numbers

Finite connected components of the aliquot graph

Conditional on a strong form of the Goldbach conjecture, we determine all finite connected components of the aliquot graph containing a number less than $10^9$, as well as those containing an amicable pair below $10^{14}$ or one of the known perfect or sociable cycles below $10^{17}$. Along the way we develop a fast algorithm for computing the inverse image of an even number under the sum-of-proper-divisors function.

On amicable tuples

For an integer $k\ge 2$, a tuple of $k$ positive integers $(M_{i})_{i=1}^{k}$ is called an amicable $k$-tuple if the equation \begin{equation*}\sigma (M_{1})=\cdots =\sigma (M_{k})=M_{1}+\cdots +M_{k}\end{equation*} holds. This is a generalization of amicable pairs. An amicable pair is a pair of distinct positive integers each of which is the sum of the proper divisors of the other. Gmelin (Über vollkommene und befreundete Zahlen (1917) Heidelberg University) conjectured that there is no relatively prime amicable pairs and Artjuhov (Acta Arith. 27 (1975) 281–291) and Borho (Math. Ann. 209 (1974) 183–193) proved that for any fixed positive integer $K$, there are only finitely many relatively prime amicable pairs $(M,N)$ with $\omega (MN)=K$. Recently, Pollack (Mosc. J. Comb. Number Theory 5 (2015), 36–51) obtained an upper bound \begin{equation*}MN<(2K)^{2^{K^{2}}}\end{equation*} for such amicable pairs. In this paper, we improve this upper bound to \begin{equation*}MN<\frac{\pi^{2}}{6}2^{4^{K}-2\cdot 2^{K}}\end{equation*} and generalize this bound to some class of general amicable tuples.

An asymptotic for the average number of amicable pairs for elliptic curves

AbstractAmicable pairs for a fixed elliptic curve defined over ℚ were first considered by Silverman and Stange where they conjectured an order of magnitude for the function that counts such amicable pairs. This was later refined by Jones to give a precise asymptotic constant. The author previously proved an upper bound for the average number of amicable pairs over the family of all elliptic curves. In this paper we improve this result to an asymptotic for the average number of amicable pairs for a family of elliptic curves.

On the numbers of prime factors of square free amicable pairs

On the numbers of prime factors of square free amicable pairs

Amicable pairs and aliquot cycles for elliptic curves over number fields

Let $E/\mathbb{Q} $ be an elliptic curve. Silverman and Stange define primes $p$ and $q$ to be an elliptic, amicable pair if $\#E(\mathbb{F} _p) = q$ and $\#E(\mathbb{F} _q) = p$. 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 $(\mathfrak{p} _1, \ldots , \mathfrak{p} _{n})$ where the $\mathfrak{p} _{i}$ are primes of~$K$ satisfying mild conditions.

Sums of Euler products and statistics of elliptic curves

We present several results related to statistics for elliptic curves over a finite field $\mathbb{F}_p$ as corollaries of a general theorem about averages of Euler products that we demonstrate. In this general framework, we can reprove known results such as the average Lang-Trotter conjecture, the average Koblitz conjecture, and the vertical Sato-Tate conjecture, even for very short intervals, not accessible by previous methods. We also compute statistics for new questions, such as the problem of amicable pairs and aliquot cycles, first introduced by Silverman and Stange. Our technique is rather flexible and should be easily applicable to a wide range of similar problems. The starting point of our results is a theorem of Gekeler which gives a reinterpretation of Deuring's theorem in terms of an Euler product involving random matrices, thus making a direct connection between the (conjectural) horizontal distributions and the vertical distributions. Our main technical result then shows that, under certain conditions, a weighted average of Euler products is asymptotic to the Euler product of the average factors.

Amicable pairs with few distinct prime factors

An amicable pair [Formula: see text] is a pair of distinct positive integers [Formula: see text] such that [Formula: see text], where [Formula: see text] is the sum of divisors function. In this note, we prove that if [Formula: see text] are amicable and [Formula: see text] is odd, then [Formula: see text], where [Formula: see text] is the number of distinct prime factors.

A note on amicable numbers and their variations

For any positive integer n, let \(\sigma (\mathrm{n})\) and p(n) denote the sum of divisors and the least prime divisor of n respectively. Let a, b be positive integers. In this paper we prove the following two results: (i) If 4 | a and \(\gcd (a, b)=1\), then a and b do not satisfy \(\sigma (a)= \sigma (b)=a+b\). (ii) If \(a>10^{8}\) and \(p(a)>2\log _{2}a+1\), where \(\log _{2}{a}\) is the logarithm of a with base 2, then a and b do not satisfy \(\sigma (a)=\sigma (b)=a+b+\lambda \), where \(\lambda \in \{0,\pm 1\}\).

Amicable pairs and aliquot cycles on average

Silverman and Stange defined the notion of an aliquot cycle of length L for a fixed elliptic curve E/ℚ, and conjectured an order of magnitude for the function that counts such aliquot cycles. We show that the conjectured upper bound holds for the number of aliquot cycles on average over the family of all elliptic curves with short bounds on the size of the parameters in the family.

Harmonious pairs

Let σ be the usual sum-of-divisors function. We say that a and b form a harmonious pair if [Formula: see text]; equivalently, the harmonic mean of [Formula: see text] and [Formula: see text] is 2. For example, 4 and 12 form a harmonious pair, since [Formula: see text] and [Formula: see text]. Every amicable pair is harmonious, but there are many others. We show that the count of numbers that belong to a harmonious pair having both members in [1, x] is at most [Formula: see text], as x → ∞.

Of love and fur: grandmothers, class, and pre-mortem inheritance in a southern Polish city

This article analyses the pre-mortem alternate-generation inheritance of fur clothing in Krakow: a practice that speaks to the conventions of grandmother-granddaughter relationships and also to the connection between inheritance and social class. The perception of fur as a symbolically loaded material enhances its suitability for being ‘kept in the family’. Clothing inheritance fosters bourgeois belonging over multiple generations, encouraging the patronage of selected spaces and services (including furriers). In the shorter term, however, generational differences within amicable grandmother-granddaughter pairs are amplified in part through the way pre-mortem inheritance practices highlight granddaughters' non-reciprocable debt to their grandmothers. A focus on fur as a particular kind of object highlights the significance of certain inheritance practices and their links to the affective ties in the reproduction of class in Poland. The article also contributes to a broader understanding of the reproduction of class by analysing the interconnections between taste, the body, and memory.

English

English

Gaussian Amicable Pairs

This article defines amicable pairs in the complex numbers and finds that some amicable pairs in the natural numbers are also amicable in the complex numbers. Unlike the case in the natural numbers, it is proved that no $(2,1)$ pairs made up of natural numbers where the common factor is a power of $2$ exist as Gaussian amicable pairs. Many pairs are found with complex parts using the DivisorSigma function in Mathematica. The factorizations into primes is given so that the type of pair might be determined.

On the distribution of some integers related to perfect and amicable numbers

Let s′(n) = ∑ d|n, 1<d<n d be the sum of the nontrivial divisors of the natural number n, where nontrivial excludes both 1 and n. For example, s′(20) = 2 + 4 + 5 + 10 = 21. A natural number n is called quasiperfect if s′(n) = n, while n and m are said to form a quasiamicable pair if s′(n) = m and s′(m) = n; in the latter case, both n and m are called quasiamicable numbers. In this paper, we prove two statistical theorems about these classes of numbers. First, we show that the count of quasiperfect n ≤ x is at most x 1 4, as x → ∞. In fact, we show that for each fixed a, there are at most x 1 4+o(1) natural numbers n ≤ x with σ(n) ≡ a (mod n) and σ(n) odd. (Quasiperfect n satisfy these conditions with a = 1.) For fixed δ 6= 0, define the arithmetic function sδ(n) := σ(n)−n− δ. Thus, s1 = s′. Our second theorem says that the number of n ≤ x which are amicable with respect to sδ is at most x/(log x) .

Morphisms preserving the set of words coding three interval exchange

Any amicable pair \phi, \psi{} of Sturmian morphisms enables a construction of a ternary morphism \eta{} which preserves the set of infinite words coding 3-interval exchange. We determine the number of amicable pairs with the same incidence matrix in $SL^\pm(2,N)$ and we study incidence matrices associated with the corresponding ternary morphisms \eta.

Amicable Pairs and Aliquot Cycles for Elliptic Curves

An amicable pair for an elliptic curve is a pair of primes (p, q) of good reduction for E and . 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 aliquot cycles, but that CM elliptic curves (with j≠0) have no aliquot cycles of length greater than two. We give conjectural formulas for the frequency of amicable pairs. For CM curves, the derivation of precise conjectural formulas involves a detailed analysis of the values of the Grössencharacter evaluated at primes in having the property that is prime. This is especially intricate for the family of curves with j=0.

Powerful amicable numbers

Let s(n) := ∑ d|n, d<n d denote the sum of the proper divisors of the natural number n. Two distinct positive integers n and m are said to form an amicable pair if s(n) = m and s(m) = n; in this case, both n and m are called amicable numbers. The first example of an amicable pair, known already to the ancients, is {220, 284}. We do not know if there are infinitely many amicable pairs. In the opposite direction, Erdős showed in 1955 that the set of amicable numbers has asymptotic density zero. Let ` ≥ 1. A natural number n is said to be `-full (or `-powerful) if p divides n whenever the prime p divides n. As shown by Erdős and Szekeres in 1935, the number of `-full n ≤ x is asymptotically c`x, as x→∞. Here c` is a positive constant depending on `. We show that for each fixed `, the set of amicable `-full numbers has relative density zero within the set of `-full numbers.