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A cluster prime is a prime p such that every even natural number k ≤ p − 3 is the difference of two primes not exceeding p. 3, 5, 7, 11, 13, 17, 19, 23, ... (OEIS: A038134) All odd primes between 3 and 89, inclusive, are cluster primes. The first 10 primes that are not cluster primes are: 2, 97, 127, 149, 191, 211, 223, 227, 229, 251.
As of October 2024, the seven largest known primes are Mersenne primes. [3] The last eighteen record primes were Mersenne primes. [4] [5] The binary representation of any Mersenne prime is composed of all ones, since the binary form of 2 k − 1 is simply k ones. [6] Finding larger prime numbers is sometimes presented as a means to stronger ...
If were to be considered a prime, many statements involving primes would need to be awkwardly reworded. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1 {\displaystyle 1} , because every number would have multiple factorizations with any number of ...
The table below lists the largest currently known prime numbers and probable primes (PRPs) as tracked by the PrimePages and by Henri & Renaud Lifchitz's PRP Records. Numbers with more than 2,000,000 digits are shown.
c. 20,000 BC — Nile Valley, Ishango Bone: suggested, though disputed, as the earliest reference to prime numbers as also a common number. [1] c. 3400 BC — the Sumerians invent the first so-known numeral system, [dubious – discuss] and a system of weights and measures.
See List of prime numbers for definitions and examples of many classes of primes. Pages in category "Classes of prime numbers" The following 76 pages are in this category, out of 76 total.
In 1954 Carlitz gave a simple proof of the weaker result that there are in general infinitely many irregular primes. [7] Metsänkylä proved in 1971 that for any integer T > 6, there are infinitely many irregular primes not of the form mT + 1 or mT − 1, [8] and later generalized this. [9]
However, modern mathematicians generally believe that his axioms were highly incomplete, and that his definitions were not really used in his proofs. 300 BC: Finite geometric progressions are studied by Euclid in Ptolemaic Egypt. [43] 300 BC: Euclid proves the infinitude of primes. [44] 300 BC: Euclid proves the Fundamental Theorem of Arithmetic.