Number Theory

Topological Algebras by V. K. Balachandran

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By V. K. Balachandran

This article, dedicated to normal topological algebras, offers a self-contained and systematic exposition of easy issues relating such algebras. either complicated and actual topological algebras, with and with no team spirit, were taken care of. A uncomplicated historical past in algebra, useful research and basic topology, is needed to appreciate this publication.

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Example text

For positive integers s and t, the minimal element of the set Ms ∩ Mt is called the least common multiple of s and t and is denoted by lcm(s, t). The following properties are easily obtained from the definition above: (1 ) If m = lcm(s, t), m = ss = tt , then gcd(s , t ) = 1. (2 ) If m is a common multiple of s and t and m = ss = tt , gcd(s , t ) = 1, then m = m. (3 ) If m is a common multiple of s and t, then m | m . β β (4 ) If s = p1α1 · · · pkαk and t = p1 1 · · · pk k , αi , bi ≥ 0, i = 1, .

9. Find all positive integers n, n ≥ 1, such that n 2 + 3n is a perfect square. Solution. Let m be a positive integer such that m 2 = n 2 + 3n . Since (m −n)(m +n) = 3n , there is k ≥ 0 such that m −n = 3k and m +n = 3n−k . From m − n < m + n follows k < n − k, and so n − 2k ≥ 1. If n − 2k = 1, then 2n = (m + n) − (m − n) = 3n−k − 3k = 3k (3n−2k − 1) = 3k (31 − 1) = 2 · 3k , so n = 3k = 2k + 1. We have 3m = (1 + 2)m = 1 + 2m + 22 m2 + · · · > 2m + 1. Therefore k = 0 or k = 1, and consequently n = 1 or n = 3.

Pn are distinct primes, and α1 , . . , αn , β1 , . . , βn , γ1 , . . , γn are nonnegative integers. Then n gcd(a, b) gcd(b, c) gcd(c, a) = gcd(a, b, c)2 min{αi ,βi } n min{βi ,γi } pi i=1 n pi i=1 n min{γi ,αi } pi i=1 2 min{αi ,βi ,γi } pi i=1 n min{αi ,βi }+min{βi ,γi }+min{γi ,αi }−2 min{αi ,βi ,γi } = pi i=1 and n lcm(a, b) lcm(b, c) lcm(c, a) = lcm(a, b, c)2 max{αi ,βi } n i=1 n max{βi ,γi } pi pi i=1 n i=1 2 max{αi ,βi ,γi } pi i=1 n = max{αi ,βi }+max{βi ,γi }+max{γi ,αi }−2 max{αi ,βi ,γi } pi i=1 max{γi ,αi } pi .

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