Numbers: Fun and Challenge
... 4t2 = (u − s2 )(u + s2 ), there exist positive integers a, b such that u − s2 = 2b2 , u + s2 = 2a2 , t2 = ab, gcd(a, b) = 1. From t2 = ab we see that there exist integers x1 , y1 such that a = x21 , b = y12 and t = x1 y1 . It follows that u = x41 + y14 and s2 = x41 − y14 . Let z1 = s, so (x1 , y1 , ...
... 4t2 = (u − s2 )(u + s2 ), there exist positive integers a, b such that u − s2 = 2b2 , u + s2 = 2a2 , t2 = ab, gcd(a, b) = 1. From t2 = ab we see that there exist integers x1 , y1 such that a = x21 , b = y12 and t = x1 y1 . It follows that u = x41 + y14 and s2 = x41 − y14 . Let z1 = s, so (x1 , y1 , ...
Fermat’s Last Theorem can Decode Nazi military Ciphers
... equation from the 6th century B.C. to measure a triangle in the history of mathematics which was Pythagoras’ geometric theorem a^2+b^2=c^2 as his premise. After Fermat developed his equation he began to substitute the exponents from 3 to 4,5,6 on up, and ...
... equation from the 6th century B.C. to measure a triangle in the history of mathematics which was Pythagoras’ geometric theorem a^2+b^2=c^2 as his premise. After Fermat developed his equation he began to substitute the exponents from 3 to 4,5,6 on up, and ...
Fermat Numbers in the Pascal Triangle
... The above result can be interpreted by saying that the Fermat numbers sit in the Pascal triangle only in the trivial way. A connection of the Fermat numbers with the Pascal triangle was pointed out in the paper of Hewgill [1]. We mention that several other diophantine equations involving the Pascal ...
... The above result can be interpreted by saying that the Fermat numbers sit in the Pascal triangle only in the trivial way. A connection of the Fermat numbers with the Pascal triangle was pointed out in the paper of Hewgill [1]. We mention that several other diophantine equations involving the Pascal ...
Review of divisibility and primes
... Definition 1. Given integers a, b, we say a divides b, or that a is a divisor of b, and write a|b, if there exists an integer q such that b = q · a. Definition 2. An integer a > 1 is prime if its only positive integer divisors are 1 and itself. Definition 3. Given a, b ∈ Z not both zero, the greates ...
... Definition 1. Given integers a, b, we say a divides b, or that a is a divisor of b, and write a|b, if there exists an integer q such that b = q · a. Definition 2. An integer a > 1 is prime if its only positive integer divisors are 1 and itself. Definition 3. Given a, b ∈ Z not both zero, the greates ...
THE FERMAT EQUATION 1. Fermat`s Last Theorem for n = 4 The proof
... Principal Ideal Domain. The proof is similar to that for the Gaussian integers – we show that any element of the quotient field Q[ζ6 ] differs from an element of Z[ζ6 ] by a complex number of norm less than 1. But completing the proof given this knowledge is not at all straightforward – it would pro ...
... Principal Ideal Domain. The proof is similar to that for the Gaussian integers – we show that any element of the quotient field Q[ζ6 ] differs from an element of Z[ζ6 ] by a complex number of norm less than 1. But completing the proof given this knowledge is not at all straightforward – it would pro ...
Number theory
Number theory (or arithmetic) is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called ""The Queen of Mathematics"" because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers).Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter (Diophantine approximation).The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by ""number theory"". (The word ""arithmetic"" is used by the general public to mean ""elementary calculations""; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.