# Opinion – Marcelo Viana: All numbers are made of primes

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The statement and proof have already appeared in Euclid’s “Elements” (c. 300 BC) but are certainly even older: every whole number greater than 1 can be expressed as a product of (one or more) prime numbers, and that expression , called factorization, is unique if we fix the order of factors. For example, the factorization of 60 is 2x2x3x5. The factorization of a prime number, such as 17, consists only of itself.

This means that primes are the pieces that all numbers are made from, rather like chemical elements make up all substances. It is one of the most important theorems in mathematics and has numerous consequences. Also because it is not just a privilege of whole numbers.

In his great work “Disquisitiones arithmeticae”, published in 1801, Carl Friedrich Gauss (1777–1855) used this theorem to prove the law of quadratic reciprocity, a profound fact of number theory that Gauss called his “golden theorem”.

Three decades later, he returned to the subject, introducing numbers of the form m+n√-1 with men integers, which we now call Gaussian integers. He proved that they also have the unique factorization property, and used this fact to prove the law of biquadratic reciprocity (stronger than quadratic).

But the unique factorization property is not always true! For example, in the set of numbers of the form m+n√-5, the number 6 can be factored into primes in two different ways: 6=2×3=(1+√-5)x(1-√-5).

To solve this problem, in 1847 the German mathematician Ernst Eduard Kummer (1810–1893) introduced a generalization of the prime idea called the ideal number, which allowed us to obtain a general single factorization theorem for all sets of numbers. The theory of ideals became one of the foundations of modern abstract algebra.

The most spectacular application was in the study of “Fermat’s Last Theorem”, which says that the equation xn+ andn= zn it has no positive integer solutions if the exponent n is greater than 2. Kummer realized that the error of many previous attempts to prove this fact lay in the misuse of single factorization, and used his ideas to correct the strategy.

In this way, he proved that the theorem holds for every exponent n that is a regular prime (most primes are believed to be regular). This was the biggest advance in this area until 1993–94, when Briton Andrew Wiles gave a complete proof of Fermat’s Last Theorem to all exponents.

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