In this paper we determine the structure of the group of all operations that send each Legendre pair to an equivalent Legendre pair.

We show in each of the Tribonacci, Padovan, Van der Laan, Perrin, Leonardo, and Narayana's cows sequences that there are infinitely many terms that cannot be represented as the sum of two prime powers.

F. Luca proved that for any fixed rational number alpha > 0 the Diophantine equations of the form alpha m! = f(n!), where f is either the Euler function or the divisor sum function or the function counting the number of divisors, have only finitely many integer solutions (m,n). In this paper we generalize that result and show that Diophantine equations of the form alpha m(1)! m(r)! = f(n!) have finitely many integer solutions, too. In addition, we include the case where f is the sum-of-kth-powers-of-divisors function. Moreover, the same holds on replacing some of the factorials with certain Bhargava factorials.

We describe how previously known methods for determining the number of decimation classes of density $\delta$ binary vectors can be extended to nonnegative integer vectors, where the vectors are indexed by a finite abelian group $G$ of size $\ell$ and exponent $\ell^*$ such that $\delta$ is relatively prime to $\ell^*$. We extend the previously discovered theory of multipliers for arbitrary subsets of finite abelian groups, to arbitrary multisets of finite abelian groups. Moreover, this developed theory provides information on the number of distinct translates fixed by each member of the multiplier group as well as sufficient conditions for each member of the multiplier group to be translate fixing.

We present a method for determining the number of decimation classes of density $$\delta $$ binary vectors indexed by a finite abelian group G of size $$\ell $$ and exponent $${\ell ^*}$$ such that $$\delta $$ is relatively prime to $${\ell ^*}$$ . This method is the first which is not based on exhaustive vector generation and exploits the subgroup lattice of $${{\mathbb {Z}}_{{\ell ^*}}^{\times }}$$ . Instead, our method is based on our newly developed theory of multipliers for arbitrary subsets of finite abelian groups, our results on orbits under the action of the multiplier group, and finding the number of solutions of a potentially highly symmetric subset sum problem. Implementing our method on vectors indexed by $${\mathbb {Z}}_{\ell }$$ of odd length $$\ell $$ and density $$(\ell +1)/2$$ greatly increased the number of $$\ell $$ for which the number of decimation classes of such vectors is known. Additionally, our newly developed theory provides information on the number of distinct translates fixed by each member of the multiplier group as well as sufficient conditions for each member of the multiplier group to be translate fixing.

It is known that there are infinitely many Sierpiński numbers and Riesel numbers in the sequences of triangular numbers, hexagonal numbers, pentagonal numbers, and many other polygonal sequences. Let Tk denote the kth triangular number. We prove an additional property: for infinitely many k, every integer in the sequence Tk2 + 1 with n a positive integer always has at least two distinct prime divisors. Furthermore, there are infinitely many k such that every integer in the sequence Tk2 1 with n a positive integer always has at least two distinct prime divisors. Also, there are infinitely many k such that every integer in both sequences Tk2 +1 and Tk2 1 with n a positive integer always has at least two distinct prime divisors. Moreover, the above results hold when replacing Tk with infinitely many di↵erent s-gonal number sequences.

In this paper, we show that there are infinitely many Sierpinski numbers in the sequence of triangular numbers, hexagonal numbers, and pentagonal numbers. We also show that there are infinitely many Riesel numbers in the same sequences. Furthermore, we show that there are infinitely many n-gonal numbers that are simultaneously Sierpinski and Riesel.

In this paper, we show that there are infinitely many Sierpiński numbers in the sequence of Lucas numbers. We also show that there are infinitely many Riesel numbers in the sequence of Lucas numbers. Finally, we show that there are infinitely many Lucas numbers that are not a sum of two prime powers.

For f one of the classical arithmetic functions d, ϕ and σ, we establish constraints on the quadruples (n, m, a, b) of integers satisfying f(n!)/m! = a/b. In particular, our results imply that as nm tends to infinity, the number of distinct prime divisors dividing the product of the numerator and denominator of the fraction f(n!)/m!, when reduced, tends to infinity.

For f one of the classical arithmetic functions d, phi and sigma, we establish constraints on the quadruples (n, m, a, b) of integers satisfying f(n!)/m! = a/b. In particular, our results imply that as nm tends to infinity, the number of distinct prime divisors dividing the product of the numerator and denominator of the fraction f(n!)/m!, when reduced, tends to infinity.

1 Waring’s Problem Problem 1. (Waring’s) For every natural number k ≥ 2 there exists a positive integer s such that every natural number is the sum of at most s k powers of natural number (for example, every natural number is the sum of at most 4 squares, or 9 cubes, or 19 fourth powers, etc.). The affirmative answer, known as the Hilbert-Waring theorem, was provided by Hilbert in 1909. Let A = (am) denote a strictly increasing sequences of nonnegative integers. Consider F (z) = ∞ ∑

We examine and classify the solutions to certain Diophantine equations involving factorials and some well known arithmetic functions. F. Luca has showed that there are finitely many solutions to the equation: f(n!) = a · m! where f is one of the arithmetic functions or (sum of the divisors function) and a is a rational number. We study the solutions for this equation when a is a prime power or a reciprocal of a prime power. Furthermore, we prove that if % is prime and k > 0, then (n!) = % k · m! and % k · f(n!) = m! have finitely many solutions (%,k,m,n), too.