Papers by Joshua Zelinsky
arXiv (Cornell University), Feb 21, 2024
Write T (n) as the sum of the reciprocals of the primes which divide n. Write H(n) = p|n p/(p − 1... more Write T (n) as the sum of the reciprocals of the primes which divide n. Write H(n) = p|n p/(p − 1) where the product is over the prime divisors of n. We prove new bounds for T (n) and H(n) in terms of the smallest prime factor of n, under the assumption that n is an odd perfect number. Some of the results also apply under the weaker assumption that n is odd and primitive nondeficient.
arXiv (Cornell University), Dec 17, 2023
We use the weighted version of the arithmetic-mean-geometric-mean inequality to motivate new resu... more We use the weighted version of the arithmetic-mean-geometric-mean inequality to motivate new results about Zaremba's function, z(n) = d|n log d d. We investigate record-setting values for z(n) and the related function v(n) = z(n) log τ (n) where τ (n) is the number of divisors of n. We show that v(n) takes on a maximum value and we give a list of all record-setting values forv(n). Closely connected inequalities motivate the study of numbers which are pseudoperfect in a strong sense.
arXiv (Cornell University), Oct 9, 2023
In this note, we fix a gap in a proof of the first author that 28 is the only even perfect number... more In this note, we fix a gap in a proof of the first author that 28 is the only even perfect number which is the sum of two perfect cubes. We also discuss the situation for higher powers.
Acta Arithmetica, 2016
We present upper bounds on certain sums which are related to Artin's primitive root conjecture an... more We present upper bounds on certain sums which are related to Artin's primitive root conjecture and are also used in counting ray class characters.
International Journal of Number Theory, Jul 1, 2019
Acquaah and Konyagin showed that if N is an odd perfect number with prime factorization N = p a 1... more Acquaah and Konyagin showed that if N is an odd perfect number with prime factorization N = p a 1 1 p a 2 2 • • • p a k k where p 1 < p 2 • • • < 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 then these bounds can be further strengthened.
arXiv (Cornell University), Oct 30, 2018
Let N be an odd perfect number. Let ω(N) be the number of distinct prime factors of N and let Ω(N... more 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 302 113 ω− 286 113 ≤ Ω. If 3|N , then 66 25 ω − 5 ≤ Ω. 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. In this paper we prove Theorem 2. If 3 |N, then Ω(N) ≥ 302 113 ω(N) − 286 113 (5) If 3|N, then Ω(N) ≥ 66 25 ω(N) − 5 (6) Note that while Inequality 6 is always better than Inequality 4, Inequality 5 is only better than Inequality 3 when ω ≥ 34. Note that the worst case of the above is when 3|N, and so we have Corollary 3. If N is an odd perfect number then Ω(N) ≥ 66 25 ω(N) − 5. Note that Kevin Hare [17] has shown that in general any odd perfect number must satisfy Ω ≥ 75, while [13] has improved this to Ω ≥ 101. This paper will contain seven sections. The first section contains various results we will need to prove Theorem 2. The second section contains the proof of Theorem 2 when 3|N. The third section contains the proof when (3, N) = 1. The fourth section improves on the known lower bound of Ω in terms of the smallest prime factor of N. This is essentially a small improvement of existing results although new questions are raised based on some aspects of the methods used. The fifth section combines the ideas of the previous sections to improve lower bounds for N in terms of its smallest prime factor. The sixth section discusses a new way of measuring how strong a statement is about odd perfect numbers and evaluates the Ochem and Rao type bounds in that context. The seventh section discusses various related open problems that are naturally connected to improving these results. We will use the following notation: N will be an odd perfect number. We will write Ω for Ω(N) and write ω for ω(N). We recall Euler's classical theorem on odd perfect numbers. Euler proved that N must have the form N = q e m 2 where q is a prime such that q ≡ e ≡ 1 (mod 4) and (q, m) = 1. Traditionally q is called the special prime. 1 Note that from Euler's result one immediately has Ω ≥ 2ω − 1. Essentially all improvements on Ochem-Rao type inequalities can be thought of as improving on the bound one has from Euler's theorem. For the remainder of this paper we will assume that
arXiv (Cornell University), May 25, 2020
Let σ(n) be the sum of the positive divisors of n. A number n is non-deficient if σ(n) ≥ 2n. We e... more Let σ(n) be the sum of the positive divisors of n. A number n is non-deficient if σ(n) ≥ 2n. We establish new lower bounds for the number of distinct prime factors of an odd non-deficient number in terms of its second smallest, third smallest and fourth smallest prime factors. We also obtain tighter bounds for odd perfect numbers. We also discuss the behavior of σ(n!+1), σ(2 n +1), and related sequences.
arXiv (Cornell University), Nov 5, 2022
Define n to be the complexity of n, which is the smallest number of 1s needed to write n using an... more Define n to be the complexity of n, which is the smallest number of 1s needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that n ≥ 3 log 3 n for all n. Richard Guy noted the trivial upper bound that n ≤ 3 log 2 n for all n > 1 by writing n in base 2. An upper bound for almost all n was provided by Juan Arias de Reyna and Jan Van de Lune. This paper provides the first non-trivial upper bound for all n. In particular, for all n > 1 we have n ≤ A log n where A =
arXiv (Cornell University), Jul 8, 2013
We present upper bounds on certain sums which are related to Artin's primitive root conjecture an... more We present upper bounds on certain sums which are related to Artin's primitive root conjecture and are also used in counting ray class characters.
Journal of Integer Sequences, Dec 1, 2002
A positive n is called a tau number if τ (n) | n, where τ is the number-ofdivisors function. Colt... more A positive n is called a tau number if τ (n) | n, where τ is the number-ofdivisors function. Colton conjectured that the number of tau numbers ≤ n is at least 1 2 π(n). In this paper I show that Colton's conjecture is true for all sufficiently large n. I also prove various other results about tau numbers and their generalizations .
Zenodo (CERN European Organization for Nuclear Research), Feb 9, 2023
Let σ(n) be the sum of the positive divisors of n. A number n is non-deficient if σ(n) ≥ 2n. We e... more Let σ(n) be the sum of the positive divisors of n. A number n is non-deficient if σ(n) ≥ 2n. We establish new lower bounds for the number of distinct prime factors of an odd non-deficient number in terms of its second smallest, third smallest and fourth smallest prime factors. Tighter bounds are obtained for odd perfect numbers. We also discuss the behavior of σ(n! + 1), σ(2 n + 1), and related sequences.
arXiv (Cornell University), Aug 26, 2019
Let N be an odd perfect number and let a be its third largest prime divisor, b be the second larg... more 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 nontrivial upper bound on a, as well as the closely related problem of improving bounds for 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 1 6. Second, we show that abc < (2N) 3 5. We also show how in certain circumstances these bounds and related inequalities can be tightened. Define a σ m,n pair to be a pair of primes p and q where q|σ(p m) and p|σ(q n). Many of our results revolve around understanding σ 2,2 pairs. We also prove results concerning σ m,n pairs for other values of m and n.
Integers, 2012
Define n to be the complexity of n, the smallest number of 1's needed to write n using an arbitra... more Define n to be the complexity of n, the smallest number of 1's needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that n ≥ 3 log 3 n for all n. Define the defect of n, denoted δ(n), to be n − 3 log 3 n; in this paper we present a method for classifying all n with δ(n) < r for a given r. From this, we derive several consequences. We prove that 2 m 3 k = 2m + 3k for m ≤ 21 with m and k not both zero, and present a method that can, with more computation, potentially prove the same for larger m. Furthermore, defining Ar(x) to be the number of n with δ(n) < r and n ≤ x, we prove that Ar(x) = Θr((log x) ⌊r⌋+1), allowing us to conclude that the values of n − 3 log 3 n can be arbitrarily large.
arXiv (Cornell University), Oct 27, 2018
Acquaah and Konyagin showed that if N is an odd perfect number with prime factorization N = p a 1... more Acquaah and Konyagin showed that if N is an odd perfect number with prime factorization N = p a 1 1 p a 2 2 • • • p a k k where p 1 < p 2 • • • < 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 then these bounds can be further strengthened.
arXiv (Cornell University), Jun 21, 2017
Let Ω(n) denote the total number of prime divisors of n (counting multiplicity) and let ω(n) deno... more 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 |N , then Ω ≥ 8 3 ω(N) − 7 3 and if 3|N then Ω(N) ≥ 21 8 ω(N) − 39 8 .
Integers, 2018
Let Ω(n) denote the total number of prime divisors of n (counting multiplicity) and let ω(n) deno... more 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 |N , then Ω ≥ 8 3 ω(N) − 7 3 and if 3|N then Ω(N) ≥ 21 8 ω(N) − 39 8 .
arXiv (Cornell University), May 25, 2020
Let n be a primitive non-deficient number where n = p a 1 1 p a 2 2 • • • p a k k where p1, p2 • ... more Let n be a primitive non-deficient number where n = p a 1 1 p a 2 2 • • • p a k k where p1, p2 • • • p k are distinct primes. We prove that there exists an i such that p a i +1 i < 2k(p1p2p3 • • • p k). We conjecture that in fact one can always find an i such that pi a i +1 < p1p2p3 • • • p k .
Cornell University - arXiv, Nov 5, 2022
Define n to be the complexity of n, which is the smallest number of 1s needed to write n using an... more Define n to be the complexity of n, which is the smallest number of 1s needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that n ≥ 3 log 3 n for all n. Richard Guy noted the trivial upper bound that n ≤ 3 log 2 n for all n > 1 by writing n in base 2. An upper bound for almost all n was provided by Juan Arias de Reyna and Jan Van de Lune. This paper provides the first non-trivial upper bound for all n. In particular, for all n > 1 we have n ≤ A log n where A =
Let N be an odd perfect number. Let ω(N) be the number of distinct prime factors of N and let Ω(N... more 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 302 113ω(N)− 286 113 ≤ Ω(N). If 3 | N , then 66 25ω(N)−5 ≤ Ω(N). 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.
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Papers by Joshua Zelinsky