Search references for LIST OF-MERSENNE-PRIMES-AND-PERFECT-NUMBERS. Phrases containing LIST OF-MERSENNE-PRIMES-AND-PERFECT-NUMBERS
See searches and references containing LIST OF-MERSENNE-PRIMES-AND-PERFECT-NUMBERS!LIST OF-MERSENNE-PRIMES-AND-PERFECT-NUMBERS
Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin
List of Mersenne primes and perfect numbers
List_of_Mersenne_primes_and_perfect_numbers
Prime number of the form 2^n – 1
definition of the Mersenne primes is that they are the prime numbers of the form Mp = 2p − 1 for some prime p. The exponents n that give Mersenne primes are
Mersenne_prime
Mathematical conjectures about Mersenne primes
the Mersenne conjectures concern the characterization of a kind of prime numbers called Mersenne primes, meaning prime numbers that are a power of two
Mersenne_conjectures
137,449,562,111 (≈6.19×1026) is the tenth Mersenne prime. See List of Mersenne primes and perfect numbers. (1000000000000000000000000000; 10009; short
Orders_of_magnitude_(numbers)
Number equal to the sum of its proper divisors
called weird numbers. Harmonic divisor number Hyperperfect number Leinster group List of Mersenne primes and perfect numbers Multiply perfect number Superperfect
Perfect_number
general one. As of October 2024[update], the seven largest known primes are Mersenne primes. The last 18 record primes were Mersenne primes. The binary representation
Largest_known_prime_number
Positive integer of the form (2^(2^n))+1
If 2k + 1 is prime and k > 0, then k itself must be a power of 2, so 2k + 1 is a Fermat number; such primes are called Fermat primes. As of 2026[update]
Fermat_number
sum of the first 4 prime numbers, and the only prime which is the sum of 4 consecutive primes. 24, all Dirichlet characters mod n are real if and only
List_of_numbers
Number of form 2^(2^p-1)-1 with prime exponent
"Martian prime". Cunningham chain Double exponential function Fermat number Perfect number Wieferich prime Chris Caldwell, Mersenne Primes: History,
Double_Mersenne_number
Number divisible only by 1 and itself
available for numbers of special forms, such as Mersenne primes, and these have been used to find large prime numbers. There are infinitely many primes, as demonstrated
Prime_number
Natural number
Logarithms and Generalized Gaussian-Mersenne Primes" (PDF). Sloane, N. J. A. (ed.). "Sequence A007770 (Happy numbers)". The On-Line Encyclopedia of Integer
23_(number)
Natural number
(Mersenne primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. "List of known Mersenne prime numbers - PrimeNet". www.mersenne.org
8191_(number)
Natural number
2147483647 is the eighth Mersenne prime, equal to 231 − 1. It is one of only four known double Mersenne primes. The primality of this number was proven
2,147,483,647
Natural number
A000043 : Mersenne exponents". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30. "Mersenne Primes: History, Theorems and Lists"
61_(number)
Are there infinitely many Kynea primes? Are there infinitely many Lucas primes? Are there infinitely many Mersenne primes (Lenstra–Pomerance–Wagstaff conjecture);
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Prime such that p^2 divides 2^(p-1)-1
including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from
Wieferich_prime
Product of an integer with itself
integer can be written as the sum of four or fewer perfect squares. Three squares are not sufficient for numbers of the form 4k(8m + 7). A positive integer
Square_number
Number raised to the third power
most numbers are not perfect cubes because all perfect cubes must have digital root 1, 8 or 9. That is their values modulo 9 may be only 0, 1, and 8. Moreover
Cube_(algebra)
Natural number with a decimal representation made of repeated instances of the same digit
the Mersenne numbers and the binary repunit primes are the Mersenne primes. It is unknown whether there are infinitely many Brazilian primes. If the Bateman–Horn
Repdigit
Natural number
prime, 127 is related to the perfect number 8128. 127 is also the largest known Mersenne prime exponent for a Mersenne number, 2 127 − 1 {\displaystyle
127_(number)
Type of prime number
≥ 1 {\displaystyle n\geq 1} and prime p ≥ n {\displaystyle p\geq n} . Generalized Wilson primes of order n are the primes p such that p 2 {\displaystyle
Wilson_prime
Numbers obtained by adding the two previous ones
can be multiply perfect, and no ratio of two Fibonacci numbers can be perfect. With the exceptions of 1, 8 and 144 (F1 = F2, F6 and F12) every Fibonacci
Fibonacci_sequence
Number used to approximate the square root of 2
equally distant from the origin and form uniform angles. A Pell prime is a Pell number that is prime. The first few Pell primes are 2, 5, 29, 5741, 33461,
Pell_number
Number of the form (n * 2^n) - 1
only if 2m + m is prime. As of January 2019, the only known primes that are both Woodall primes and Mersenne primes are W2 = M3 = 7, and W512 = M521. Like
Woodall_number
Infinite integer series where the next number is the sum of the two preceding it
and the closely related Fibonacci sequence. Individual numbers in the Lucas sequence are known as Lucas numbers. Lucas numbers and Fibonacci numbers form
Lucas_number
Mathematical concept
namely 2, 3, 5, 11, 17 and 41 (sequence A014556 in the OEIS). Note that these numbers are all prime numbers. The primes of the form k2 − k + 41 are
Lucky_numbers_of_Euler
Odd number with specific properties
Riesel Primes PrimeGrid The Riesel Problem: Definition and Status The Prime Glossary: Riesel number List of primes of the form: k*2^n-1, k<300 List of primes
Riesel_number
Prime number of the form (2ᵖ+1)/3
the prime pages credit François Morain for naming them in a lecture at the Eurocrypt 1990 conference. Wagstaff primes appear in the New Mersenne conjecture
Wagstaff_prime
Number of the form x^y + y^x
double-covering the set of Leyland numbers (so we have 1 < y ≤ x). A Leyland prime is a Leyland number that is prime. The first such primes are: 17, 593, 32993, 2097593
Leyland_number
Two raised to an integer power
Proposition 36 of Elements proves that if the sum of the first n terms of this progression is a prime number (and thus is a Mersenne prime as mentioned
Power_of_two
Natural number
2032, and 4064 add up to 8128), and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime 127, 27
8128
This is a list of notable integer sequences with links to their entries in the On-Line Encyclopedia of Integer Sequences. OEIS core sequences Index to
List_of_integer_sequences
Integer whose multiples are digit rotations
decimal), and p is a prime that does not divide b. (Primes p that give cyclic numbers in base b are called full reptend primes or long primes in base b). For
Cyclic_number
Used to count, measure, and label
formula as for odd and even numbers to generate the prime numbers. A special class are the Mersenne primes, which are prime numbers of the form 2n − 1,
Number
Natural number
that does not yield a Mersenne prime because 211 − 1 is not prime: 211 − 1 = 2047 = 23 × 89. (5, 11) is a pair of brown numbers because 5! + 1 = 112.
11_(number)
Natural number
approximation of π, 3.1415..., and a very rough approximation of e, 2.71828... 3 is the first Mersenne prime. It is also the first of five known Fermat primes. It
3
Italian mathematician (1548–1626)
the eighth Mersenne prime. Although Cataldi incorrectly claimed that p=23, 29, 31 and 37 all also generate Mersenne primes (and perfect numbers), when in
Pietro_Cataldi
Characterization of even perfect numbers
relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form 2p−1(2p − 1), where 2p − 1 is a prime number
Euclid–Euler_theorem
Type of composite number
mathematics All known Giuga numbers are even. If an odd Giuga number exists, it must be the product of at least 14 primes. It is not known if there are
Giuga_number
Integer whose representation contains every digit in its number base
except for 0 and numbers of the form 2 n − 1 {\displaystyle 2^{n}-1} (the Mersenne numbers). The larger the base, the rarer pandigital numbers become, though
Pandigital_number
Numbers whose aliquot sums form a cyclic sequence
sociable numbers are numbers whose aliquot sums form a periodic sequence. They are generalizations of the concepts of perfect numbers and amicable numbers. The
Sociable_number
Natural number
791,750 = The sum of the first 500 squared numbers 43,050,817 = Leyland number using 3 & 16 (316 + 163) 43,112,609 = Mersenne prime exponent 43,443,858
10,000,000
Natural number
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-08-29. Caldwell, Chris K. "Mersenne Primes: History, Theorems and Lists". "The Science
77_(number)
Integer filtered out using a sieve similar to that of Eratosthenes
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
Lucky_number
Number whose square ends in the same digits
f(x)=x^{2}} , and 0 and 1 are automorphic numbers in every base. These solutions are called trivial automorphic numbers. If b {\displaystyle b} is a prime power
Automorphic_number
Number that cannot be written as an aliquot sum
sum of its own proper divisors. Similarly, none of the amicable numbers or sociable numbers are untouchable. Also, none of the Mersenne numbers are untouchable
Untouchable_number
Natural number
centered heptagonal number 3937 – product of distinct Mersenne primes, repeated sum of divisors is prime, denominator of conversion factor from meter to US survey
3000_(number)
Pair of integers related by their divisors
cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs. Amicable numbers are featured in the book The Man
Amicable_numbers
Natural number
Germain prime. 2005 = 5 × 401. It is a vertically symmetric number. 2011 is a sexy prime with 2017. It is also the sum of eleven consecutive primes: 2011
2000_(number)
Number that is the sum of its iterated totients
2 and relatively prime to it, so the totient of 2 is 1; and 9 = 6 + 2 + 1, so 9 is a perfect totient number. The first few perfect totient numbers are
Perfect_totient_number
Recursive integer sequence
Catalan's triangle Catalan–Mersenne number Delannoy number Fuss–Catalan number List of factorial and binomial topics Lobb numbers Motzkin number Narayana
Catalan_number
Numbers with many divisors
prime numbers pi must be precisely the first k prime numbers (2, 3, 5, ...); if not, we could replace one of the given primes by a smaller prime, and
Highly_composite_number
Positive integer of the form 4n + 1
sequence of Hilbert primes begins 5, 9, 13, 17, 21, 29, 33, 37, 41, 49, ... (sequence A057948 in the OEIS). A Hilbert prime is not necessarily a prime number;
Hilbert_number
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A000668 (Mersenne primes)". The On-Line Encyclopedia of Integer Sequences
1,000,000,000
Composite number that passes Fermat's probable primality test
product of consecutive Fermat numbers is a base-2 pseudoprime, and so are all Fermat composites and Mersenne composites. The probability of a composite
Fermat_pseudoprime
pseudoprime Probable prime Baillie–PSW primality test Miller–Rabin primality test Lucas–Lehmer primality test Lucas–Lehmer test for Mersenne numbers AKS primality
List_of_number_theory_topics
if and only if the real part of s is greater than 1. The sum of the reciprocals of all the Mersenne numbers is Erdős–Borwein constant. The sum of the
List_of_sums_of_reciprocals
Numbers parameterizing ways to partition a set
Stirling numbers of the second kind occur in combinatorics and the study of partitions. They are named after James Stirling. The Stirling numbers of the first
Stirling numbers of the second kind
Stirling_numbers_of_the_second_kind
Natural number
sum of the squares of the first fourteen primes 8269 – cuban prime of the form x = y + 1 8273 – Sophie Germain prime 8281 = 912, sum of the cubes of the
8000_(number)
Natural number
and preceding 1024. 1023 is a number whose sum of digits is 6, and is a sum of 5 consecutive primes (193, 197, 199, 211 and 223). It is a sub-perfect
1023_(number)
Product of numbers from 1 to n
squarefree. As with the factorial primes n ! ± 1 {\displaystyle n!\pm 1} , researchers have studied primorial primes n # ± 1 {\displaystyle n\#\pm 1}
Factorial
Mathematical concept in prime numbers
set containing infinitely many primes and missing infinitely many other primes. A positive integer n is idoneal if and only if it cannot be written as
Idoneal_number
Number that is less than the sum of its proper divisors
k primes (sequence A047802 in the OEIS). If A ( k ) {\displaystyle A(k)} represents the smallest abundant number not divisible by the first k primes then
Abundant_number
Mathematical sequences in combinatorics
In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in
Stirling_number
Class of natural numbers with many divisors
positive integer. The first ten superior highly composite numbers and their factorization are listed. For a superior highly composite number n there exists
Superior highly composite number
Superior_highly_composite_number
Number of dots in a centred dot square
} That is, the nth centered square number is the sum of the nth and the (n − 1)th square numbers. The following pattern demonstrates this formula: The
Centered_square_number
Concept in number theory
obvious factor of 5). The general form of the factorization was later discovered by Lucas. 536903681 is an example of a Gaussian Mersenne norm. A. Granville
Aurifeuillean_factorization
Number that remains the same when its digits are reversed
(sequence A006995 in the OEIS) The Fermat primes and the Mersenne primes form a subset of the binary palindromic primes. Any number n {\displaystyle n} is palindromic
Palindromic_number
Centered figurate number that represents a nonagon with a dot in the center
numbers 28 and 496. All even perfect numbers are triangular numbers whose index is an odd Mersenne prime. Since every Mersenne prime greater than 3 is congruent
Centered_nonagonal_number
Numbers that evenly divide powers of 60
Regular numbers are numbers that evenly divide powers of 60 (or, equivalently, powers of 30). Equivalently, they are the numbers whose only prime divisors
Regular_number
Integer of the form 3 × 2^n – 1 for non-negative n
n+2 digits long, consisting of "10" followed by n 1s. The first few Thabit numbers that are prime (Thabit primes or 321 primes): 2, 5, 11, 23, 47, 191, 383
Thabit_number
Mathematical concept
any sequence of numbers n·2n + a + b where a and b are integers, and in particular also for Woodall numbers. The only known Cullen primes, sixteen in all
Cullen_number
divisor number Sphenic number Smith number Double Mersenne number Zeisel number Heteromecic number Niven numbers Superparticular number Highly composite number
List of recreational number theory topics
List_of_recreational_number_theory_topics
Numbers whose prime factors all divide the number more than once
squareful, which refers to numbers that are not square-free.) The following is a list of all powerful numbers between 1 and 1000: 1, 4, 8, 9, 16, 25, 27
Powerful_number
Special type of prime number
these primes first arose due to their connection with Fermat's Last Theorem. Wolstenholme primes are also related to other special classes of numbers, studied
Wolstenholme_prime
Product of the first "n" prime numbers
the sequence of the prime numbers. Griffiths (2015) proved that it is irrational. Euclid's proof of his theorem on the infinitude of primes can be paraphrased
Primorial
Type of natural number
hyperperfect numbers", Abstracts of the American Mathematical Society, 1 (6): 561. Minoli, Daniel; Nakamine, W. (1980). "Mersenne numbers rooted on 3 for
Hyperperfect_number
Arithmetic operation
involving two numbers: the base, b, and the exponent, n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base:
Exponentiation
Concatenation of the first n prime numbers
result of concatenating the first 128 prime numbers, through 719. The primes at the end of the concatenation in the Smarandache–Wellin primes are 2, 3
Smarandache–Wellin_number
Natural number
= 28 – 1, it is a Mersenne number (though not a pernicious one), and the fifth such number not to be a prime number. It is a perfect totient number, the
255_(number)
Repeated sum of a number's digits
digital roots are the natural numbers 0 ≤ n < b {\displaystyle 0\leq n<b} , and there are no cycles other than the fixed points of 0 ≤ n < b {\displaystyle
Digital_root
1955 mathematics book by Constance Reid
of computers to discover Mersenne primes. She published an article on a closely related topic, perfect numbers, in Scientific American in 1953, and wrote
From_Zero_to_Infinity
Number that is the sum of its own digits, each raised to a given power
number. The perfect digital invariant function (also known as a happy function, from happy numbers) for base b > 1 {\displaystyle b>1} and power p > 0
Perfect_digital_invariant
Munchausen number
In number theory, a perfect digit-to-digit invariant (PDDI; also known as a Munchausen number) is a natural number in a given number base b {\displaystyle
Perfect digit-to-digit invariant
Perfect_digit-to-digit_invariant
Composite number in number theory
Carmichael numbers are odd, since any even composite number that is square-free (and hence has only one prime factor of two) will have at least one odd prime factor
Carmichael_number
Square of a triangular number
of the odd numbers, the first is the cube of 1, the sum of the next two is the cube of 2, the sum of the next three is the cube of 3, and so on. He does
Squared_triangular_number
Positive integer whose divisors have a harmonic mean that is an integer
But no perfect number can be a square: this follows from the known form of even perfect numbers and from the fact that odd perfect numbers (if they
Harmonic_divisor_number
Ten raised to an integer power
magnitude (numbers) for named powers of ten. There are two conventions for naming positive powers of ten, beginning with 109, called the long and short scales
Power_of_10
Mathematics book written in 1202 by Fibonacci
remainder theorem, perfect numbers and Mersenne primes as well as formulas for arithmetic series and for square pyramidal numbers. Another example in
Liber_Abaci
Number sequence 3,0,2,3,2,5,5,7,10,...
final state of the above six registers (the "signature" of n) equals the initial state 1,−1,3, 3,0,2. The same occurs with ≈ 1/6 of all primes, so the two
Perrin_number
Composite number with special property
needed] The set of all n-Knödel numbers is denoted Kn. The special case K1 is the Carmichael numbers. There are infinitely many n-Knödel numbers for a given
Knödel_number
French mathematician and lawyer (1601–1665)
Pell's equation, perfect numbers, amicable numbers and what would later become Fermat numbers. It was while researching perfect numbers that he discovered
Pierre_de_Fermat
of the set of algebraic numbers ( ℵ 0 {\displaystyle \aleph _{0}} ). Bernhard Riemann, at the end of his famous 1859 paper "On the Number of Primes Less
List_of_conjectures
Type of positive integer
numbers between 2184 and 2200 = 2184 + 16 each share a prime factor with one of 2184 = 23 · 3 · 7 · 13 and 2200 = 23 · 52 · 11. These 15 numbers and their
Erdős–Woods_number
Integer that occurs often as a totient
highly totient numbers get tougher to find the higher one goes, since calculating the totient function involves factorization into primes, something that
Highly_totient_number
Natural number
(43411) and 20 (16120). It is a Mersenne exponent, i.e. 2521−1 is prime. It is the largest known such exponent that is the lesser of twin primes. 4521 -
500_(number)
Series of books
of Square, Triangle, Cube, Centred Hexagon and Tetrahedral numbers, "difference of two squares", number superstitions, prime numbers, Mersenne primes
Murderous_Maths
Mathematical recursive sequence
aliquot sequences eventually end with a prime number, a perfect number, or a set of amicable or sociable numbers? (Catalan's aliquot sequence conjecture)
Aliquot_sequence
Number used for counting
number as a product of primes Countable set – Mathematical set that can be enumerated Sequence – Function of the natural numbers in another set Ordinal
Natural_number
Numeral ambigram
strobogrammatic in binary. Dihedral primes that do not use 2 or 5 are also strobogrammatic primes in binary. The natural numbers 0 and 1 are strobogrammatic in every
Strobogrammatic_number
travel, tourism, insurance
LIST OF-MERSENNE-PRIMES-AND-PERFECT-NUMBERS
LIST OF-MERSENNE-PRIMES-AND-PERFECT-NUMBERS
LIST OF-MERSENNE-PRIMES-AND-PERFECT-NUMBERS
LIST OF-MERSENNE-PRIMES-AND-PERFECT-NUMBERS
LIST OF-MERSENNE-PRIMES-AND-PERFECT-NUMBERS
LIST OF-MERSENNE-PRIMES-AND-PERFECT-NUMBERS
LIST OF-MERSENNE-PRIMES-AND-PERFECT-NUMBERS
LIST OF-MERSENNE-PRIMES-AND-PERFECT-NUMBERS
LIST OF-MERSENNE-PRIMES-AND-PERFECT-NUMBERS
travel, tourism, insurance