Prime Numbers

Since all other whole numbers (except 0) can be produced by multiplying together primes – they must be considered fundamental.

(1), 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 &etc

There are an infinite number of primes - as proved by Euclid around 300B.C. ( Ref. ). Although billions of them have so far been found (nowadays via computational 'sieving' algorithms), they appear to be randomly distributed - but with some 'rules' - for example, they become more rare as the numbers get bigger. As yet there’s no proven mathematical method which can accurately predict where the next one will be.

Note: As at Oct. 2024. the largest known prime number (discovered via collaborative networked computer processing) has 41,024,320 digits (ref.)

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Also see: The Riemann hypothesisplugin-autotooltip__plain plugin-autotooltip_bigThe Riemann hypothesis

- was proposed by Bernhard Riemann (1859), and is a conjecture about the distribution of the zeros of the Riemann zeta function.

The Riemann hypothesis asserts that all interesting solutions of the equation :

$$ {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}},} $$

lie on a certain vertical straight line. and Legendre's Conjectureplugin-autotooltip__plain plugin-autotooltip_bigLegendre's Conjecture

Legendre's Conjecture concerns the distribution of Prime Numbers

It asks : "Is there is a prime number between n2 and (n + 1)2 for every positive integer n. squares?"

It was first presented by French mathematician Adrien-Marie Legendre in the early 1800s - and to date has neither been proved or disproved. and Oppermann's conjectureplugin-autotooltip__plain plugin-autotooltip_bigOppermann's conjecture

Oppermann's Conjecture concerns the distribution of Prime Numbers.

It was first suggested in 1877, and has not been proved or disproved since then.

The conjecture states that : for every integer x > 1, there is at least one prime number between

Note: The true nature of randomnessplugin-autotooltip__plain plugin-autotooltip_bigRandom numbers

unknowable

"We can never decide for sure that a number is random, but what we can do is apply an increasing number of tests and treat our sequence of numbers as innocent until proved guilty."

Source : Prof. Colva Roney-Dougal, senior lecturer in Pure Mathematics at the University of St Andrews, speaking in itself is also under discussion.

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