A rationality condition is derived for the existence of odd perfect numbers involving the square root of a product, which consists of a sequence of repunits, multiplied by twice the base of one of the repunits. This constraint also provides an upper bound for density of odd integers which could satisfy ${{σ(N)}\over N}=2$, where $N$ belongs to a fixed interval with a lower limit greater than $10^{300}$. Characteristics of prime divisors of repunits are used to establish whether the product containing the repunits can be a perfect square. It is shown that the arithmetic primitive divisors with different prime bases can be equal only when the exponents are different, with the exception of a set of cases derived from solutions of a solution prime equation. The proof of this result requires the demonstration of the non-existence of solutions of a more general prime equation, a problem which is equivalent to Catalan's conjecture. Results concerning the exponents of prime divisors of the repunits are obtained, and they are combined with the method of induction to prove to prove a general theorem on the non-existence of prime divisors satisfying the rationality condition.
We call $n$ a spoof odd perfect number if $n$ is odd and $n=km$ for two integers $k,m>1$ such that $σ(k)(m+1)=2n$, where $σ$ is the sum-of-divisors function. In this paper, we show how results analogous to those of odd perfect numbers could be established for spoof odd perfect numbers (otherwise known in the literature as Descartes numbers). In particular, we prove that $k$ is not almost perfect.
We give necessary conditions satisfied by the set of odd prime divisors of binary perfect polynomials. This allows us to get a new characterization of all the known perfect binary polynomials.
An odd perfect number, N, is shown to have at least nine distinct prime factors. If 3 does not divide N, then N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.
We present lower bounds on the sum and product of the distinct prime factors of an odd perfect number, which provide a lower bound on the size of the odd perfect number as a function of the number of its distinct prime factors.
We extend the sum-of-divisors function to the complex plane via the Gaussian integers. Then we prove a modified form of Euler's classification of odd perfect numbers.
Let $k\ge2$ be an integer. A natural number $n$ is called $k$-perfect if $σ(n)=kn.$ For any integer $r\ge1$ we prove that the number of odd $k$-perfect numbers with at most $r$ distinct prime factors is bounded by $k4^{r^3}$.
We investigate the integer solutions of Diophantine equations related to perfect numbers. These solutions generalize the example, found by Descartes in 1638, of an odd, ``spoof'' perfect factorization $3^2\cdot 7^2\cdot 11^2\cdot 13^2\cdot 22021^1$. More recently, Voight found the spoof perfect factorization $3^4\cdot 7^2\cdot 11^2\cdot 19^2\cdot(-127)^1$. No other examples appear in the literature. We compute all nontrivial, odd, primitive spoof perfect factorizations with fewer than seven bases -- there are twenty-one in total. We show that the structure of odd, spoof perfect factorizations …
Much recent progress has been made concerning the probable existence of Odd Perfect Numbers, forming part of what has come to be known as Sylvester's Web Of Conditions. This paper proves some results concerning certain properties of the sums of reciprocals of the factors of odd perfect numbers, or, in more technical terms, the properties of the sub-sums of σ_{-1} (n). By this result, it also establishes strong bounds on the prime factors of odd perfect numbers using the number of distinct prime factors it may possess.
The existence of a perfect odd number is an old open problem of number theory. An Euler's theorem states that if an odd integer $ n $ is perfect, then $ n $ is written as $ n = p ^ rm ^ 2 $, where $ r, m $ are odd numbers, $ p $ is a prime number of the form $ 4 k + 1 $ and $ (p, m) = 1 $, where $ (x, y) $ denotes the greatest common divisor of $ x $ and $ y $. In this article we show that the exponent $ r $, of $ p $, in this equation, is necessarily equal to 1. That is, if $ n $ is an odd perfect number, then $ n $ is written as $ n = pm ^ 2. $
We study the set $\mathcal{S}$ of odd positive integers $n$ with the property ${2n}/{σ(n)} - 1 = 1/x$, for positive integer $x$, i.e., the set that relates to odd perfect and odd "spoof perfect" numbers. As a consequence, we find that if $D=pq$ denotes a spoof odd perfect number other than Descartes' example, with pseudo-prime factor $p$, then $q>10^{12}$. Furthermore, we find irregularities in the ending digits of integers $n\in\mathcal{S}$ and study aspects of its density, leading us to conjecture that the amount of numbers in $\mathcal{S}$ below $k$ is $\sim10 \log(k)$.
For a positive integer $n$, let $σ(n)$ denote the sum of the positive divisors of $n$. Let $d$ be a proper divisor of $n$. We call $n$ a deficient-perfect number if $σ(n)=2n-d$. In this paper, we show that the only odd deficient-perfect number with four distinct prime divisors is $3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2}$.
The only (unitary) perfect polynomials over $\mathbb{F}_2$ that are products of $x$, $x+1$ and Mersenne primes are precisely the nine (resp. nine "classes") known ones. This follows from a new result about the factorization of $M^{2h+1} +1$, for a Mersenne prime $M$ and for a positive integer $h$. Other consequences of such a factorization are new results about odd perfect polynomials.
The Descartes-Frenicle-Sorli conjecture predicts that $k=1$ if $q^k n^2$ is an odd perfect number with Euler prime $q$. In this note, we present some conditions equivalent to this conjecture.
The Euler's form of odd perfect numbers, if any, is $n=π^αN^2$, where $π$ is prime, $(π,N)=1$ and $π\equiv α\equiv 1 \pmod{4}$. Dris conjecture states that $N>π^α$. We find that $N^2>\frac{1}{2}π^γ$, with $γ=max\{ω(n)-1,α\}$; $ω(n)\geq 9$ is the number of distinct prime factors of $n$.
If $N={q^k}{n^2}$ is an odd perfect number given in Eulerian form, then the Descartes-Frenicle-Sorli conjecture predicts that $k=ν_{q}(N)=1$. In this article, we give a short proof for this conjecture.
Euler showed that if an odd perfect number exists, it must be of the form $N = p^αq_{1}^{2β_{1}}$ $\ldots$ $q_{k}^{2β_{k}}$, where $p, q_{1}, \ldots, q_k$ are distinct odd primes, $α$, $β_{i} \geq 1$, for $1 \leq i \leq k$, with $p \equiv α\equiv 1 \pmod{4}$. In 2005, Evans and Pearlman showed that $N$ is not perfect, if $3|N$ or $7|N$ and each $β_{i} \equiv 2 \pmod{5}$. We improve on this result by removing the hypothesis that $3|N$ or $7|N$ and show that $N$ is not perfect, simply, if each $β_{i} \equiv 2 \pmod{5}$.
Let $N$ be an odd perfect number. Let $ω(N)$ be the number of distinct prime factors of $N$ and let $Ω(N)$ be the total number (counting multiplicity) of prime factors of $N$. We prove that $\frac{99}{37}ω(N) - \frac{187}{37} \leq Ω(N)$ and that if $3\nmid N$, then $\frac{51}{19}ω(N)-\frac{46}{19} \leq Ω(N)$.
We shall show that, for any given primes $\ell\geq 17$ and $p, q\equiv 1\pmod{\ell}$, the diophantine equation $(x^\ell-1)/(x-1)=p^m q$ has at most four positive integral solutions $(x, m)$ and give its application to odd perfect number problem.
We shall given a new effectively computable upper bound of odd perfect numbers whose Euler factors are powers of fixed exponent, improving our old result in T. Yamada, Colloq. Math. 103 (2005), 303--307.
A natural number $n$ is called {\it multiperfect} or {\it$k$-perfect} for integer $k\ge2$ if $σ(n)=kn$, where $σ(n)$ is the sum of the positive divisors of $n$. In this paper, we establish the structure theorem of odd multiperfect numbers analogous as Euler's theorem on odd perfect numbers. We prove the divisibility of the Euler part of odd multiperfect numbers and characterize the forms of odd perfect numbers $n=π^αM^2$ such that $π\equivα(\text{mod}8)$. We also present some examples to show the nonexistence of odd perfect numbers as applications.
We investigate the implications of a curious biconditional involving divisors of odd perfect numbers, if Dris conjecture that $q^k < n$ holds, where $q^k n^2$ is an odd perfect number with Euler prime $q$. We then show that this biconditional holds unconditionally. Lastly, we prove that the inequality $q<n$ holds unconditionally.
In this note, we show that if $N$ is an odd perfect number and $q^α$ is some prime power exactly dividing it, then $σ(N/q^α)/q^α>5$. In general, we also show that if $σ(N/q^α)/q^α<K$, where $K$ is any constant, then $N$ is bounded by some function depending on $K$.
If $N={q^k}{n^2}$ is an odd perfect number given in Eulerian form, then the Descartes-Frenicle-Sorli conjecture predicts that $k=1$. Brown has recently announced a proof for the inequality $q < n$, and a partial proof that $q^k < n$ holds under many cases. In this article, we give a strategy for strengthening Brown's result to $q^2 < n$.
If $N = {q^k}{n^2}$ is an odd perfect number, where $q$ is the Euler prime, then we show that $n < q$ is sufficient for Sorli's conjecture that $k = ν_{q}(N) = 1$ to hold. We also prove that $q^k < 2/3{n^2}$, and that $I(q^k) < I(n)$, where $I(x)$ is the abundancy index of $x$.
Let $p^k m^2$ be an odd perfect number with special prime $p$. In this article, we provide an alternative proof for the biconditional that $σ(m^2) \equiv 1 \pmod 4$ holds if and only if $p \equiv k \pmod 8$. We then give an application of this result to the case when $σ(m^2)/p^k$ is a square.
Let $Ω(n)$ denote the total number of prime divisors of $n$ (counting multiplicity) and let $ω(n)$ denote the number of distinct prime divisors of $n$. Various inequalities have been proved relating $ω(N)$ and $Ω(N)$ when $N$ is an odd perfect number. We improve on these inequalities. In particular, we show that if $3 \not| N$, then $Ω\geq \frac{8}{3}ω(N)-\frac{7}{3}$ and if $3 |N$ then $Ω(N) \geq \frac{21}{8}ω(N)-\frac{39}{8}.$
If $N = {p^k}{m^2}$ is an odd perfect number with special prime factor $p$, then it is proved that ${p^k} < (2/3){m^2}$. Numerical results on the abundancy indices $\frac{σ(p^k)}{p^k}$ and $\frac{σ(m^2)}{m^2}$, and the ratios $\frac{σ(p^k)}{m^2}$ and $\frac{σ(m^2)}{p^k}$, are used. It is also showed that $m^2 > \frac{\sqrt{6}}{2}({10}^{150})$.
Let $σ(n)$ denote the sum of the positive divisors of $n$. We say that $n$ is perfect if $σ(n) = 2 n$. Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form $N = p^α\prod_{j=1}^k q_j^{2 β_j}$, where $p, q_1, ..., q_k$ are distinct primes and $p \equiv α\equiv 1 \pmod{4}$. Define the total number of prime factors of $N$ as $Ω(N) := α+ 2 \sum_{j=1}^k β_j$. Sayers showed that $Ω(N) \geq 29$. This was later extended by Iannucci and Sorli to show that $Ω(N) \geq 37$. This was extended by the author to show that $Ω(N) \geq 47$…
Acquaah and Konyagin showed that if $N$ is an odd perfect number where $N= p_1^{a_1}p_2^{a_2} \cdots p_k^{a_k}$ where $p_1 < p_2 \cdots < p_k$ then one must have $p_k < 3^{1/3}N^{1/3}$. Using methods similar to theirs, we show that $p_{k-1}< (2N)^{1/5}$ and that $p_{k-1}p_k < 6^{1/4}N^{1/2}.$ We also show that if $p_k$ and $p_{k-1}$ are close to each other than these bounds can be further strengthened.
We study some divisibility properties of multiperfect numbers. Our main result is: if $N=p_1^{α_1}... p_s^{α_s} q_1^{2β_1}... q_t^{2β_t}$ with $β_1, ..., β_t$ in some finite set S satisfies $σ(N)=\frac{n}{d}N$, then N has a prime factor smaller than C, where C is an effective computable constant depending only on s, n, S.
Let $p^k m^2$ be an odd perfect number with special prime $p$. Extending previous work of the authors, we prove that the inequality $m < p^k$ follows from $m^2 - p^k = 2^r t$, where $r \geq 2$ and $\gcd(2,t)=1$, under the following hypotheses: (a) $m > t > 2^r$, or (b) $m > 2^r > t$. We also prove that the estimate $m^2 - p^k > 2m$ holds. We can also improve this unconditional estimate to $m^2 - p^k > {313m^2}/315$.
Let $N$ be an odd perfect number and let $a$ be its third largest prime divisor, $b$ be the second largest prime divisor, and $c$ be its largest prime divisor. We discuss steps towards obtaining a non-trivial upper bound on $a$, as well as the closely related problem of improving bounds $bc$, and $abc$. In particular, we prove two results. First we prove a new general bound on any prime divisor of an odd perfect number and obtain as a corollary of that bound that $$a < 2N^{\frac{1}{6}}.$$ Second, we show that $$abc < (2N)^{\frac{3}{5}}.$$ We also show how in certain circumstances these bounds …
If $N={q^k}{n^2}$ is an odd perfect number given in Eulerian form, then Sorli's conjecture predicts that $k=ν_{q}(N)=1$. In this article, we give some further results related to this conjecture and those contained in the papers \cite{Dris} and \cite{Dris2}. (withdrawn because of a crucial gap in Theorem 1 [see https://arxiv.org/pdf/1309.0906.pdf for what is currently provable in this regard], as well as elementary mistakes in the numerical bounds from pages 2 to 4)
We address an arithmetic problem in the ring $\F_2[x]$ related to the fixed points of the sum of divisors function. We study some binary polynomials $A$ such that $σ(A)/A $ is still a binary polynomial. Technically, we prove that the only (unitary) perfect polynomials over $\F_2$ that are products of $x$, $x+1$ and of Mersenne primes are precisely the nine (resp. nine "classes") known ones. This follows from a new result about the factorization of $M^{2h+1} +1$, for a Mersenne prime $M$ and for a positive integer $h$.
Let $N$ be an odd perfect number. Let $ω(N)$ be the number of distinct prime factors of $N$ and let $Ω(N)$ be the total number of prime factors of $N$. We prove that if $(3,N)=1$, then $ \frac{302}{113}ω- \frac{286}{113} \leq Ω. $ If $3|N$, then $\frac{66}{25}ω-5\leqΩ.$ This is an improvement on similar prior results by the author which was an improvement of a result of Ochem and Rao. We also establish new lower bounds on $ω(N)$ in terms of the smallest prime factor of $N$ and establish new lower bounds on $N$ in terms of its smallest prime factor.
Dris conjectured in his masters thesis that the inequality $q^k < n$ always holds, if $N = {q^k}{n^2}$ is an odd perfect number with special prime $q$. In this note, we initially show that either of the two conditions $n < q^k$ or $σ(q)/n < σ(n)/q$ holds. This is achieved by first proving that $σ(q)/n \neq σ(n)/q^k$, where $σ(x)$ is the sum of the divisors of $x$. Using this analysis, we further show that the condition $q < n < q^k$ holds in four out of a total of six cases. Finally, we prove that $n < q^k$, and that this holds unconditionally. This finding disproves both the Dris Conjecture a…
Let $σ(x)$ be the sum of the divisors of $x$. If $N$ is odd and $σ(N) = 2N$, then the odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n) = 1$. In this note, we show that $q < n$ implies that Descartes's conjecture (previously Sorli's conjecture), $k = ν_{q}(N) = 1$, is not true. This then implies an unconditional proof for the biconditional $$k = ν_{q}(N) = 1 \Longleftrightarrow n < q.$$ Lastly, following a recent result of Cohen and Sorli, we show that if $q < n$, then either $q > 5$ or $k > 5$ i…
In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic $p$-rational number fields of degree $2^{t}$ for any odd prime number $p$ and any integer $t \geq 1$. Using the criteria provided by him to check $p$-rationality for abelian number fields, certain infinite families of quadratic, biquadratic and triquadratic $p$-rational fields have been shown to exist in recent years. In this article, for any integer $k \geq 1$, we build upon the existing work and prove the existence of infinitely many prime numbers $p$ for which the imaginary quadrat…
Let $q^k n^2$ be an odd perfect number with special prime $q$. Define the GCDs $$G = \gcd\bigg(σ(q^k),σ(n^2)\bigg)$$ $$H = \gcd\bigg(n^2,σ(n^2)\bigg)$$ and $$I = \gcd\bigg(n,σ(n^2)\bigg).$$ We prove that $G \times H = I^2$. (Note that it is trivial to show that $G \mid I$ and $I \mid H$ both hold.) We then compute expressions for $G, H,$ and $I$ in terms of $σ(q^k)/2, n,$ and $\gcd\bigg(σ(q^k)/2,n\bigg)$. Afterwards, we prove that if $G = H = I$, then $σ(q^k)/2$ is not squarefree. Other natural and related results are derived further. Lastly, we conjecture that the set $$\mathscr{A} = \{m : \g…
The multiplicative structure of an odd perfect number $n$, if any, is $n=π^αM^2$, where $π$ is prime, $\gcd(π,M)=1$ and $π\equiv α\equiv1\pmod{4}$. An additive structure of $n$, established by Touchard, is that "$\bigl(n\equiv 9\pmod{36}\bigr )$ OR $\bigl (n\equiv1\pmod{12}\bigr )$". A first extension of Touchard's result is that the proposition "$\bigl(n\equiv x^2\pmod{4 x^2}\bigr )$ OR $\bigl (n\equiv π\equiv1\pmod{4 x}\bigr )$" holds for $x=3$ (the extension is due to the fact that the second congruence contains also $π$). We further extend the proof to $x=α+2$, $α+2$ prime, with the restri…
We shall give an explicit upper bound for the smallest prime factor of multiperfect numbers of the form $N=p_1^{α_1}\cdots p_s^{α_s} q_1^{β_1}\cdots q_t^{β_t}$ with $β_1, \ldots, β_t$ bounded by a given constant. We shall also give similar results for quasiperfect numbers and relatively prime amicable pairs of opposite parity.
The divisor function $σ(n)$ sums the divisors of $n$. We call $n$ abundant when $σ(n) - n > n$ and perfect when $σ(n) - n = n$. I recently introduced the recursive divisor function $a(n)$, the recursive analog of the divisor function. It measures the extent to which a number is highly divisible into parts, such that the parts are highly divisible into subparts, so on. Just as the divisor function motivates the abundant and perfect numbers, the recursive divisor function motivates their recursive analogs, which I introduce here. A number is recursively abundant, or ample, if $a(n) > n$ and recu…
Let p be an odd prime, n an odd positive integer and C the p-Sylow subgroup the class group of the p-cyclotomic extension of the rationals. When log(p) is bigger than n**(224n**4), we prove that the eigenspace on C attached to the (p-n)-th power of the Teichmuller character is trivial. The proof uses the K-theory of the integers and the Voronoi reduction theory of quadratic forms.
We adapt (over $\mathbb{F}_2$) the general notions of multiplicative function, Dirichlet convolution and Inverse. We get some interesting results, namely necessary conditions for an odd binary polynomial to be perfect. Note that we are inspired by the "analogous" works in \cite{Gall-Rahav-newcongr} and \cite{Touchard}, about odd perfect numbers.
A perfect number is a positive integer $N$ such that the sum of all the positive divisors of $N$ equals $2N$, denoted by $σ(N) = 2N$. The question of the existence of odd perfect numbers (OPNs) is one of the longest unsolved problems of number theory. This thesis presents some of the old as well as new approaches to solving the OPN Problem. In particular, a conjecture predicting an injective and surjective mapping $X = σ(p^k)/p^k, Y = σ(m^2)/m^2$ between OPNs $N = {p^k}{m^2}$ (with Euler factor $p^k$) and rational points on the hyperbolic arc $XY = 2$ with $1 < X < 1.25 < 1.6 < Y < 2$ and $2.8…
Euler showed that if an odd perfect number $N$ exists, it must consist of two parts $N=q^k n^2$, with $q$ prime, $q \equiv k \equiv 1 \pmod{4}$, and gcd$(q,n)=1$. Dris conjectured that $q^k < n$. We first show that $q<n$ for all odd perfect numbers. Afterwards, we show $q^k < n$ holds in many cases.
In this note we will analyze a diophantine equation raised by Michael Bennett in [1] that is pivotal in establishing that powers of five has few digits in its ternary expansion. We will show that the Diophantine equation $3^{a}+3^{b}+2=n^5$, where $(n,3)=1$ and $a>b>0$ is insoluble for pairs of positive integers $(a,b)$ where they are both even or one is even and the other is odd. In the case where both $(a,b)$ are odd, there is one known solution $2^5=3^3+3^1+2.$ We will show that there are no other solutions to the diophantine equation for $n^{5}<32\left(1+3(10^6)\right)^5$.
The equality between the number of odd spin structures on a Riemann surface of genus g, with $2^g - 1$ being a Mersenne prime, and the even perfect numbers, is an indication that the action of the modular group on the set of spin structures has special properties related to the sequence of perfect numbers. A method for determining whether Mersenne numbers are primes is developed by using a geometrical representation of these numbers. The connection between the non-existence of finite odd perfect numbers and the irrationality of the square root of twice the product of a sequence of repunits is …
