JP Journal of Algebra, Number Theory and Applications
Volume 48, Issue 1, Pages 109 - 119
(October 2020) http://dx.doi.org/10.17654/NT048010109 |
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SQUARE-FREE INTEGERS CODIFIED INTO THE BINARY STRINGS AND THEIR ASSOCIATED GROUPS
Đặng Võ Phúc, Salman Khan and Shahid Nawaz
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Abstract: In this paper, the square-free integers are expressed as binary strings. First, the integer is written in the form of its prime factors. Then, in the prime factorization, the primes (bases) are dropped and the integer is expressed as a string of the exponents. As the integers involved are square-free, the associated strings involve binary digits. These binary strings satisfy certain product and sum rules and form groups under addition and multiplication. It does not, however, form a field as the distributive property does not hold. We also explore the algebra of infinity. It turns out that infinity plays the role of additive identity. |
Keywords and phrases: prime numbers, fundamental theorem of arithmetic, group theory, binary strings, infinity.
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