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Wednesday, November 18, 2020

The Fundamental Theorem of Arithmetic | Mathematics - Medium

Over 2,300 years ago, Euclid proved the Fundamental Theorem of Arithmetic. Now it is our turn by Maths and Musings published in Cantor’s Paradise

Some surviving fragments from Euclid’s Elements

Statement of the Theorem

Take our first prime, ‘p_1’. Clearly p_1 divides N. Now, as p_1 divides N, p_1 divides the second prime factorisation. By (repeated application of) Euclid’s Lemma, this means that p_1 divides one of the prime factors of N. But as these numbers are prime, p_1 must equal one of the prime factors. This is a contradiction, as we had already cancelled out all the shared prime factors.

Thus, the factorisation into primes must be unique.

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Source: Medium