Polynomial Prime Producers: A Computer Problem for Secondary Students

1975 ◽  
Vol 68 (2) ◽  
pp. 108-111
Author(s):  
James K. Bidwell ◽  
Ann Fallon
Keyword(s):  
Number Theory ◽  
Secondary Students ◽  
Unsolved Problem ◽  
Polynomial Function ◽  
Prime Numbers ◽  
Elementary Number Theory ◽  
Elementary Number ◽  
Computer Problem

An unsolved problem in mathematics is the existence of a formula (function) that produces nothing but prime numbers. Many mathematicians have thought that they had discovered such a function, only to have composite numbers eventually pop up for some integer substitutions. In particular, it is known that no polynomial function over the integers can produce only primes. In fact, the proof of this theorem is not difficult and is found in many elementary number theory texts.

scholarly journals Elementary Number Theory Problems. Part II

2021 ◽  
Vol 29 (1) ◽  
pp. 63-68
Author(s):  
Artur Korniłowicz ◽  
Dariusz Surowik
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Prime Numbers ◽  
Elementary Number Theory ◽  
Elementary Number ◽  
Composite Number

Summary In this paper problems 14, 15, 29, 30, 34, 78, 83, 97, and 116 from [6] are formalized, using the Mizar formalism [1], [2], [3]. Some properties related to the divisibility of prime numbers were proved. It has been shown that the equation of the form p 2 + 1 = q 2 + r 2, where p, q, r are prime numbers, has at least four solutions and it has been proved that at least five primes can be represented as the sum of two fourth powers of integers. We also proved that for at least one positive integer, the sum of the fourth powers of this number and its successor is a composite number. And finally, it has been shown that there are infinitely many odd numbers k greater than zero such that all numbers of the form 22 n + k (n = 1, 2, . . . ) are composite.

Elementary Number Theory

2016 ◽  
pp. 1-32
Author(s):  
Gary L. Mullen ◽  
James A. Sellers
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Elementary Number

Some Elementary Number Theory

2019 ◽  
pp. 239-244
Author(s):  
Richard Evan Schwartz
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Result Section ◽  
Structural Result ◽  
Elementary Number

This chapter proves some number-theoretic results about the sequences defined in Chapter 23. It proceeds as follows. Section 24.2 proves Lemma 24.1, a multipart structural result. Section 24.3 takes care of several number-theoretic details left over from Section 23.6 and Section 23.7.

Infinite Orbits Revisited

2019 ◽  
pp. 227-238
Author(s):  
Richard Evan Schwartz
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Elementary Number ◽  
Model Section

This is the first of four chapters giving a self-contained proof of Theorem 0.7. Section 23.2 describes a sequence of even rationals {pn/qn} that converges to A. Section 23.3 states the two main technical results, the Box Theorem and the Copy Theorem. Section 23.4 shows how to choose a sequence {cn}. Section 23.5 states three auxiliary results about arc copying in the plaid model. Section 23.6 deduces the Box Theorem from one of these auxiliary lemmas. Section 23.7 deduces the Copy Theorem from the auxiliary lemmas and some elementary number theory. Thus, after this chapter ends, the only remaining task is to prove the auxiliary copy lemmas and prove a few lemmas in elementary number theory.

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