17 Facts About Prime numbers

Prime numbers conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but was unable to prove it.

Prime numbers introduced methods from mathematical analysis to this area in his proofs of the infinitude of the primes and the divergence of the sum of the reciprocals of the primes.

The idea that prime numbers had few applications outside of pure mathematics was shattered in the 1970s when public-key cryptography and the RSA cryptosystem were invented, using prime numbers as their basis.

The mathematical theory of prime numbers moved forward with the Green–Tao theorem that there are arbitrarily long arithmetic progressions of prime numbers, and Yitang Zhang's 2013 proof that there exist infinitely many prime gaps of bounded size.

Some other more technical properties of prime numbers do not hold for the number 1: for instance, the formulas for Euler's totient function or for the sum of divisors function are different for prime numbers than they are for 1.

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Just as the distance between two real Prime numbers can be measured by the absolute value of their distance, the distance between two rational Prime numbers can be measured by their -adic distance, the -adic absolute value of their difference.

The local-global principle allows certain problems over the rational numbers to be solved by piecing together solutions from each of their places, again underlining the importance of primes to number theory.

Prime numbers ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry.

Early attempts to prove Fermat's Last Theorem led to Kummer's introduction of regular primes, integer prime numbers connected with the failure of unique factorization in the cyclotomic integers.

Prime numbers are used in computing for checksums, hash tables, and pseudorandom number generators.

Methods suitable for arbitrary large Prime numbers that do not depend on the size of its factors include the quadratic sieve and general number field sieve.

Prime numbers are used in pseudorandom number generators including linear congruential generators and the Mersenne Twister.

Prime numbers are of central importance to number theory but have many applications to other areas within mathematics, including abstract algebra and elementary geometry.

Beyond mathematics and computing, prime numbers have potential connections to quantum mechanics, and have been used metaphorically in the arts and literature.

Prime numbers are significant in quantum information science, thanks to mathematical structures such as mutually unbiased bases and symmetric informationally complete positive-operator-valued measures.

The French composer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena".

Prime numbers are used as a metaphor for loneliness and isolation in the Paolo Giordano novel The Solitude of Prime Numbers, in which they are portrayed as "outsiders" among integers.