A kaleidoscope of solutions for a Diophantine system

2010 ◽  
Vol 94 (529) ◽  
pp. 42-50
Author(s):  
Juan Pla
Keyword(s):  
Analogous Problem ◽  
Pell Equation ◽  
Pythagorean Triples ◽  
Diophantine System ◽  
Consecutive Integers ◽  
Image Position

A classical exercise in recreational mathematics is to find Pythagorean triples such that the legs are consecutive integers. It is equivalent to solve the Pell equation with k = 2. In this case it provides all the solutions (see [1] for details). But to obtain all the solutions of a Diophantine system in one stroke is rather exceptional. Actually this note will show that the analogous problem of finding four integers A, B, C and D such that

Prime Producing Quadratic Polynomials and Class-Numbers of Real Quadratic Fields

1990 ◽  
Vol 42 (2) ◽  
pp. 315-341 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Quadratic Polynomials ◽  
Real Quadratic Fields ◽  
Consecutive Integers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Real Quadratic

Frobenius-Rabinowitsch's theorem provides us with a necessary and sufficient condition for the class-number of a complex quadratic field with negative discriminant D to be one in terms of the primality of the values taken by the quadratic polynomial with discriminant Don consecutive integers (See [1], [7]). M. D. Hendy extended Frobenius-Rabinowitsch's result to a necessary and sufficient condition for the class-number of a complex quadratic field with discriminant D to be two in terms of the primality of the values taken by the quadratic polynomials and with discriminant D (see [2], [7]).

Multiplicative functions at consecutive integers

1986 ◽  
Vol 100 (2) ◽  
pp. 229-236 ◽  
Author(s):  
Adolf Hildebrand
Keyword(s):  
Random Sequence ◽  
Multiplicative Functions ◽  
Liouville Function ◽  
Prime Factors ◽  
Consecutive Integers ◽  
Image Position

Let λ(n) denote the Liouville function, i.e. λ(n) = 1 if n has an even number of prime factors, and λ(n) = − 1 otherwise. It is natural to expect that the sequence λ(n) (n ≥ 1) behaves like a random sequence of ± signs. In particular, it seems highly plausible that for any choice of εi = ± 1 (i = 0,…, k) we have

scholarly journals The negative Pell equation and Pythagorean triples

2000 ◽  
Vol 76 (6) ◽  
pp. 91-94 ◽  
Author(s):  
Aleksander Grytczk ◽  
Florian Luca ◽  
Marek Wójtowicz
Keyword(s):  
Pell Equation ◽  
Pythagorean Triples

The Thinking of Students: My Application of the Pythagorean Theorem

1996 ◽  
Vol 1 (10) ◽  
pp. 814-816
Author(s):  
Paul Pollack
Keyword(s):  
Pythagorean Theorem ◽  
Seventh Grade ◽  
Algebra Students ◽  
Pythagorean Triples ◽  
Consecutive Integers ◽  
Two Sides

One of my seventh-grade-algebra students, Paul Pollack, shared a discovery he had made. We had finished a unit of study on the Pythagorean theorem, including the Pythagorean triples. Paul noticed that in several triples, the hypotenuse was equal to one of the legs plus 1. For example, 3-4-5, 5-12-13, and 7-24-25 triples have two sides whose values are consecutive integers. Paul was intrigued and developed a pattern to derive triples wherein the hypotenuse and one leg differ by 1, or, for that matter, by any desired quantity. Note that these “triples” often take a little finagling to result in integers (e.g., in example 2). He found thallhe lengths of the three sides of which two are consecutive integers would fit the pattern

scholarly journals On the order of magnitude of Jacobsthal's function

1977 ◽  
Vol 20 (4) ◽  
pp. 329-331 ◽  
Author(s):  
R. C. Vaughan
Keyword(s):  
Prime Divisors ◽  
Order Of Magnitude ◽  
Consecutive Integers ◽  
Image Position

Let n be an integer with n > 1. Jacobsthal (3) defines g(n) to be the least integer so that amongst any g(n) consecutive integers a + 1, a + 2, … a + g(n) there is at least one coprime with n. In other words, ifthenIt is probably true thatwhere ω(n) denotes the number of different prime divisors of n, and Erdos (1) has pointed out that by the small sieve it is possible to show that there is a constant C such that

On Equal Products of Consecutive Integers

1970 ◽  
Vol 13 (2) ◽  
pp. 255-259 ◽  
Author(s):  
R. A. Macleod ◽  
I. Barrodale
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Generalized Equation ◽  
Algebraic Numbers ◽  
Numerical Evidence ◽  
Consecutive Integers ◽  
Integer Solutions ◽  
Image Position ◽  
Positive Integers

Using the theory of algebraic numbers, Mordell [1] has shown that the Diophantine equation1possesses only two solutions in positive integers; these are given by n = 2, m = 1, and n = 14, m = 5. We are interested in positive integer solutions to the generalized equation2and in this paper we prove for several choices of k and l that (2) has no solutions, in other cases the only solutions are given, and numerical evidence for all values of k and l for which max (k, l) ≤ 15 is also exhibited.

On Representations as a Sum of Consecutive Integers

1950 ◽  
Vol 2 ◽  
pp. 399-405 ◽  
Author(s):  
W. J. Leveque
Keyword(s):  
Real Parameter ◽  
Average Order ◽  
Consecutive Integers ◽  
Probability Integral ◽  
Image Position ◽  
Positive Integers

1. Introduction. It is the object of this paper to investigate the function γ(m), the number of representations of m in the form(1) where . It is shown that γ(m) is always equal to the number of odd divisors of m, so that for example γ(2k) = 1, this representation being the number 2k itself. From this relationship the average order of γ(m) is deduced ; this result is given in Theorem 2. By a method due to Kac [2], it is shown in §3 that the number of positive integers for which γ(m) does not exceed a rather complicated function of n and ω, a real parameter, is asymptotically nD(ω), where D(ω) is the probability integral

Euler's Criterion for Quintic Nonresidues

1985 ◽  
Vol 37 (5) ◽  
pp. 1008-1024 ◽  
Author(s):  
S. A. Katre ◽  
A. R. Rajwade
Keyword(s):  
Roots Of Unity ◽  
Diophantine System ◽  
Image Position

Let e be an integer ≧ 2, and p a prime = 1 (mod e). Euler's criterion states that for D ∊ Z,(1.1)if and only if D is an e-th power residue (mod p). If D is not an e-th power (mod p), one has(1.2)for some e-th root α(≠1) of unity (mod p). Sometimes expressions for roots of unity (mod p) can be given in terms of quadratic partitions of p. For example,(1.3)are the four distinct fourth roots of unity (mod p) for a prime p ≡ 1 (mod 4) in terms of a solution (a, b) of the diophantine system(a, b unique), whereas for p ≡ 1 (mod 3), a solution (L, M) of the systemgives(1.4)as the three distinct cuberoots of unity (mod p).

Sums of squares of integers in arithmetic progression

2011 ◽  
Vol 95 (533) ◽  
pp. 186-196
Author(s):  
Tom M. Apostol ◽  
Mamikon A. Mnatsakanian
Keyword(s):  
Arithmetic Progression ◽  
Sums Of Squares ◽  
Consecutive Integers ◽  
Image Position ◽  
The Right

The following striking identitiesare the cases n = 1,2,3,4 of a remarkable family given by G.J. Dostor [1]:where m = n(2n + 1), and n = 1, 2, … The case m = −n is trivial. If m ≠ −n there are n + 1 squares of consecutive integers on the left and n on the right. We will treat the last term (m + n)2 on the left differently, and refer to it as a transition term relating two sums of squares of n consecutive integers.

Multiplicative functions at consecutive integers. II

1988 ◽  
Vol 103 (3) ◽  
pp. 389-398 ◽  
Author(s):  
Adolf Hildebrand
Keyword(s):  
Sufficient Conditions ◽  
Mean Value ◽  
Arithmetic Functions ◽  
Multiplicative Functions ◽  
Global Behaviour ◽  
Consecutive Integers ◽  
The Mean ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Multiplicative Arithmetic

The global behaviour of multiplicative arithmetic functions has been extensively studied and is now well understood for a large class of multiplicative functions. In particular, Halász [5] completely determined the asymptotic behaviour of the meansfor multiplicative functions g satisfying |g| ≤ 1, and gave necessary and sufficient conditions for the existence of the ‘mean value’

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