In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form (p, p + 2, p + 6) or (p, p + 4, p + 6). [1] With the exceptions of (2, 3, 5) and (3, 5, 7), this is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime (except for 3 itself).
The first prime triplets (sequence A098420 in the OEIS ) are
(5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109), (107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227, 229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349, 353), (457, 461, 463), (461, 463, 467), (613, 617, 619), (641, 643, 647), (821, 823, 827), (823, 827, 829), (853, 857, 859), (857, 859, 863), (877, 881, 883), (881, 883, 887)
A prime triplet contains a single pair of:
A prime can be a member of up to three prime triplets - for example, 103 is a member of (97, 101, 103), (101, 103, 107) and (103, 107, 109). When this happens, the five involved primes form a prime quintuplet.
A prime quadruplet (p, p + 2, p + 6, p + 8) contains two overlapping prime triplets, (p, p + 2, p + 6) and (p + 2, p + 6, p + 8).
Similarly to the twin prime conjecture, it is conjectured that there are infinitely many prime triplets. The first known gigantic prime triplet was found in 2008 by Norman Luhn and François Morain. The primes are (p, p + 2, p + 6) with p = 2072644824759 × 233333 − 1. As of October 2020 [update] the largest known proven prime triplet contains primes with 20008 digits, namely the primes (p, p + 2, p + 6) with p = 4111286921397 × 266420 − 1. [2]
The Skewes number for the triplet (p, p + 2, p + 6) is 87613571, and for the triplet (p, p + 4, p + 6) it is 337867. [3]
A palindromic number is a number that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term palindromic is derived from palindrome, which refers to a word whose spelling is unchanged when its letters are reversed. The first 30 palindromic numbers are:
In number theory, a lucky number is a natural number in a set which is generated by a certain "sieve". This sieve is similar to the Sieve of Eratosthenes that generates the primes, but it eliminates numbers based on their position in the remaining set, instead of their value.
In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers.
In number theory, cousin primes are prime numbers that differ by four. Compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six.
90 (ninety) is the natural number following 89 and preceding 91.
105 is the natural number following 104 and preceding 106.
400 is the natural number following 399 and preceding 401.
700 is the natural number following 699 and preceding 701.
600 is the natural number following 599 and preceding 601.
800 is the natural number following 799 and preceding 801.
900 is the natural number following 899 and preceding 901. It is the square of 30 and the sum of Euler's totient function for the first 54 positive integers. In base 10, it is a Harshad number. It is also the first number to be the square of a sphenic number.
In number theory, a prime quadruplet is a set of four prime numbers of the form {p, p + 2, p + 6, p + 8}. This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4.
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are
In number theory, a full reptend prime, full repetend prime, proper prime or long prime in base b is an odd prime number p such that the Fermat quotient
In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading ("left") digit is successively removed, then all resulting numbers are prime. For example, 9137, since 9137, 137, 37 and 7 are all prime. Decimal representation is often assumed and always used in this article.
In mathematics, a Wieferich pair is a pair of prime numbers p and q that satisfy
In recreational number theory, a minimal prime is a prime number for which there is no shorter subsequence of its digits in a given base that form a prime. In base 10 there are exactly 26 minimal primes:
A Fortunate number, named after Reo Fortune, is the smallest integer m > 1 such that, for a given positive integer n, pn# + m is a prime number, where the primorial pn# is the product of the first n prime numbers.
A good prime is a prime number whose square is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes.