Products Of Consecutive Integers Research Articles

On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

On products of consecutive integers

On products of consecutive integers

Power values of sums of certain products of consecutive integers and related results

Let n be a non-negative integer and put pn(x)=∏i=0n(x+i). In the first part of the paper, for given n, we study the existence of integer solutions of the Diophantine equationym=pn(x)+∑i=1kpai(x), where m∈N≥2 and a1<a2<…<ak<n. This equation can be considered as a generalization of the Erdős–Selfridge Diophantine equation ym=pn(x). We present some general finiteness results concerning the integer solutions of the above equation. In particular, if n≥2 with a1≥2, then our equation has only finitely many solutions in integers. In the second part of the paper we study the equationym=∑i=1kpai(xi), for m=2,3, which can be seen as an additive version of the equation considered by Erdős and Graham. In particular, we prove that if m=2,a1=1 or m=3,a2=2, then for each k−1 tuple of positive integers (a2,…,ak) there are infinitely many solutions in integers.

Common factors among pairs of consecutive integers

In this paper, we study products of consecutive integers and prove an optimal bound on their greatest common factors. In contrast, we obtain a modest upper bound for the greatest common factor in the perfect square situation. Using this and an upper bound on the size of solutions to hyperelliptic curves, we prove a gap principle when [Formula: see text] divides [Formula: see text] with some additional restrictions. We also obtain a stronger gap principle under the abc conjecture.

Balancing numbers which are products of consecutive integers

Balancing numbers which are products of consecutive integers

On diophantine equations of the form ${({\boldsymbol x}-{\boldsymbol a}_{\bf 1})({\boldsymbol x}-{\boldsymbol a}_{\bf 2}) \ldots ({\boldsymbol x}-{\boldsymbol a}_{\boldsymbol k}) + {\boldsymbol r} = {\boldsymbol y}^{ \boldsymbol n}}$

Erdős and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x + 1)(x + 2)...(x + (m − 1)) = y n has no solutions in positive integers x,m,n where m, n > 1 and y ∈ Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0 ≤ a 1 < a 2 < ⋯ < a k are integers and, with r ∈ Q, n ≥ 3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n > 2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.

Combinatorial numbers in binary recurrences

We give several effective and explicit results concerning the values of some polynomials in binary recurrence sequences. First we provide an effective finiteness theorem for certain combinatorial numbers (binomial coefficients, products of consecutive integers, power sums, alternating power sums) in binary recurrence sequences, under some assumptions. We also give an efficient algorithm (based on genus 1 curves) for determining the values of certain degree 4 polynomials in such sequences. Finally, partly by the help of this algorithm we completely determine all combinatorial numbers of the above type for the small values of the parameter involved in the Fibonacci, Lucas, Pell and associated Pell sequences.

Correction to the Paper “Almost Perfect Powers in Products of Consecutive Integers” (Monatsh. Math. 145, 19–33 (2005))

Correction to the Paper “Almost Perfect Powers in Products of Consecutive Integers” (Monatsh. Math. 145, 19–33 (2005))

Almost Perfect Powers in Products of Consecutive Integers

The classical equations (1) and (3) have a very extensive literature. The main purpose of recent investigations is to solve these equations with as large bound for the greatest prime factor P(b) of b as possible. General elementary methods have been developed for studying (1) and (3) which, however, cannot be applied if k is small. As a generalization of previous results obtained for small values of k, we completely solve Eq. (1) for k 5, under the assumption that P(b) pk, the k-th prime (cf. Theorem 1). A similar result is established for Eq. (3) (cf. Theorem 2). In our proofs, several deep results and powerful techniques are combined from modern Diophantine analysis.

Number of prime divisors in a product of consecutive integers

Number of prime divisors in a product of consecutive integers

Products of disjoint blocks of consecutive integers which are powers

The product of consecutive integers cannot be a power (after Erdős and Selfridge), but products of disjoint blocks of consecutive integers can be powers. Even if the blocks have a fixed length $l\geq 4$ there are many solutions. We give the bound for the

Squares from Products of Consecutive Integers

(2002). Squares from Products of Consecutive Integers. The American Mathematical Monthly: Vol. 109, No. 5, pp. 459-462.

Squares from Products of Consecutive Integers

Squares from Products of Consecutive Integers

Almost perfect powers in consecutive integers

We study a generalized version of a theorem of Erdos and Selfridge on when a product of consecutive integers is a pure power.

Products of consecutive integers and the Markoff equation

Recently, Katayama described all integral solutions to the diophantine equationX(X + 1)Y(Y + 1) = Z(Z + 1). In this paper, we clarify his description by noting a bijection withx 2 + y 2 + z 2 = 2xyz + 5, which has been studied by Mordell. We also show the number of positive integer solutions withZ ≤H to Katayama's equation is of order\(\sqrt H \), and in general, count the number of solutions to Mordell's equationx 2 + y 2 + z 2 = axyz + b with height less thanH.

Proof without Words Sum of Products of Consecutive Integers

Proof without Words Sum of Products of Consecutive Integers

Proof without Words: Sum of Products of Consecutive Integers

Proof without Words: Sum of Products of Consecutive Integers

Products of consecutive integers and the Markoff equation

Products of consecutive integers and the Markoff equation

On products of consecutive integers

On products of consecutive integers

How Many Pairs of Products of Consecutive Integers Have the Same Prime Factors?

How Many Pairs of Products of Consecutive Integers Have the Same Prime Factors?