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Prime number that is product of first n primes ± 1
mathematics, a primorial prime is a prime number of the form pn# ± 1, where pn# is the primorial of pn (i.e. the product of the first n primes). Primality
Primorial_prime
Product of the first "n" prime numbers
the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial"
Primorial
Euclid primes are primes p such that p−1 is a primorial. 3, 7, 31, 211, 2311, 200560490131 (OEIS: A018239) Euler irregular primes are primes p {\displaystyle
List_of_prime_numbers
Number divisible only by 1 and itself
product of the prime numbers up to n {\displaystyle n} , and a primorial prime is a prime of one of the forms n # ± 1 {\displaystyle n\#\pm 1} . Gardiner
Prime_number
BOINC based volunteer computing project researching prime numbers
23# · n is prime for n = 0, ..., 26. 23# = 2·3·5·7·11·13·17·19·23 = 223092870, or 23 primorial, is the product of all primes up to 23. PrimeGrid is also
PrimeGrid
Product of prime numbers, plus one
integers of the form En = pn # + 1, where pn # is the nth primorial (the product of the first n prime numbers). They are named after the ancient Greek mathematician
Euclid_number
Prime number one less or more than a factorial
(see prime gap). Primorial prime Weisstein, Eric W. "Factorial Prime". MathWorld. The Top Twenty: Factorial primes from the Prime Pages Factorial Prime Search
Factorial_prime
Formula whose values are the prime numbers
{\displaystyle p_{n}\#} , the primorial of p n {\displaystyle p_{n}} . This formula should be seen as a recurrence relation for the prime numbers, expressing p
Formula_for_primes
Inequality relating the primorial to square of the next prime number
named after H. Bonse, relates the size of a primorial to the smallest prime that does not appear in its prime factorization. It states that for all n ≥
Bonse's_inequality
Prime number of the form 2^n – 1
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some
Mersenne_prime
Numbers with many divisors
the primorial a 0 a 1 ⋯ a n {\displaystyle a_{0}a_{1}\cdots a_{n}} . Roughly speaking, for a number to be highly composite it has to have prime factors
Highly_composite_number
Set of prime numbers linked by a linear relationship
{\displaystyle k} does not begin with the prime k {\displaystyle k} , then the common difference is a multiple of the primorial k # = 2 ⋅ 3 ⋅ 5 ⋯ j {\displaystyle
Primes in arithmetic progression
Primes_in_arithmetic_progression
Prime such that p^2 divides 2^(p-1)-1
In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem
Wieferich_prime
Natural number
– primorial prime, twin prime with 2311, Mertens function zero, highly cototient number 2310 – fifth primorial 2311 – primorial prime, twin prime with
2000_(number)
Natural number
001. 30029 = primorial prime 30030 = primorial 30031 = smallest composite number which is one more than a primorial 30203 = safe prime 30240 = harmonic
30,000
Natural number
with 31. 29 is the fifth primorial prime, the sixth Sophie Germain prime, a Lucas prime, a Pell prime, and an Eisenstein prime with no imaginary part and
29_(number)
Repeatable pattern of differences between prime numbers
multiple of the primorial of k. The Skewes numbers for prime k-tuples are an extension of the definition of Skewes's number to prime k-tuples based on
Prime_k-tuple
Integer filtered out using a sieve similar to that of Eratosthenes
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
Lucky_number
Numbers with a certain property involving recursive summation
{\displaystyle b} -happy prime will not necessarily create another happy prime. For instance, while 19 is a 10-happy prime, 91 = 13 × 7 is not prime (but is still
Happy_number
2299 (sequence A006145 in the OEIS). A primorial p n # {\displaystyle p_{n}\#} is the product of all primes from 2 to p n {\displaystyle p_{n}} . The
Table_of_prime_factors
Integer named after Reo Fortune
given positive integer n, pn# + m is a prime number, where the primorial pn# is the product of the first n prime numbers. For example, to find the seventh
Fortunate_number
Chris Caldwell, The Top Twenty: Primorial at The Prime Pages. Caldwell, Chris, "The largest known primes", The Prime Pages, archived from the original
Orders_of_magnitude_(numbers)
Product of numbers from 1 to n
number counts the symmetries of some tree. Primorial The primorial n # {\displaystyle n\#} is the product of prime numbers less than or equal to n {\displaystyle
Factorial
Natural number
000 integers 509,203 = Riesel prime 510,510 = the product of the first seven prime numbers, thus the seventh primorial. It is also the product of four
100,000
primality test for Mersenne numbers. “!” is the factorial, “#” is the primorial, and Φ 3 ( x ) {\displaystyle \Phi _{3}(x)} is the third cyclotomic polynomial
List of largest known primes and probable primes
List_of_largest_known_primes_and_probable_primes
Newman–Shanks–Williams prime Primorial prime Wagstaff prime Wall–Sun–Sun prime Wieferich prime Wilson prime Wolstenholme prime Woodall prime Prime pages Covering
List_of_number_theory_topics
Natural number
be swapped but turning over is not allowed 200,560,490,130 = eleventh primorial 208,023,278,209 = 28th Motzkin number. 225,851,433,717 = 56th Fibonacci
100,000,000,000
Prime number of the form 2^u × 3^v + 1
In number theory, a Pierpont prime is a prime number of the form 2 u ⋅ 3 v + 1 {\displaystyle 2^{u}\cdot 3^{v}+1\,} for some nonnegative integers u and
Pierpont_prime
Prime number of the form that allows fast modular reduction
In mathematics, a Solinas prime, or generalized Mersenne prime, is a prime number that has the form f ( 2 m ) {\displaystyle f(2^{m})} , where f ( x )
Solinas_prime
Natural number
number 7,163,627,708,162 : 172nd Markov number 7,420,738,134,810 : 12th primorial 7,625,597,484,987 = 19,6833 = 279 = 327 = 333 = 33 = 23, megafugathree
1,000,000,000,000
Concatenation of the first n prime numbers
Smarandache–Wellin number is a prime with 5719 digits ending in 11927, discovered by Eric W. Weisstein as a probable prime in 1998 and then proven prime in 2022. In March
Smarandache–Wellin_number
Type of prime number
In number theory, a Wilson prime is a prime number p {\displaystyle p} such that p 2 {\displaystyle p^{2}} divides ( p − 1 ) ! + 1 {\displaystyle (p-1)
Wilson_prime
Type of prime number
A cuban prime is a prime number that is also a solution to one of two different specific equations involving differences between third powers of two integers
Cuban_prime
Product of two prime numbers
product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there
Semiprime
Number of form 2^(2^p-1)-1 with prime exponent
number that is prime is called a double Mersenne prime. Since a Mersenne number Mp can be prime only if p is prime, (see Mersenne prime for a proof), a
Double_Mersenne_number
Numbers k where x - phi(x) = k has many solutions
least one prime factor in common with x {\displaystyle x} . For example, the cototient of 6 is 4 since these four positive integers have a prime factor in
Highly_cototient_number
Number that remains the same when its digits are reversed
to be those numbers that contain a factor of the primorial n#, where n≥13 and is the largest prime factor in the number. Fuller called these numbers
Palindromic_number
Number of prime factors of a natural number
is prime a lower bound on the value of the function is ω ( n ) = 1 {\displaystyle \omega (n)=1} . Similarly, if n {\displaystyle n} is primorial then
Prime_omega_function
Numbers that contain only the digit 1
repunit prime is a repunit that is also a prime number. Primes that are repunits in base-2 are Mersenne primes. As of May 2025, the largest known prime number
Repunit
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
Number whose sums of distinct divisors represent all smaller numbers
is clear. Each successive primorial is formed by multiplying a prime number p i {\displaystyle p_{i}} by a smaller primorial that is divisible by both
Practical_number
Type of Poulet number
get super-Poulet numbers with 3 distinct prime divisors. If you find three Poulet numbers with three common prime factors, you get a super-Poulet number
Super-Poulet_number
Prime number congruent to 1 mod 4
A Pythagorean prime is a prime number of the form 4 n + 1 {\displaystyle 4n+1} . Pythagorean primes are exactly the odd prime numbers that are the sum
Pythagorean_prime
Power of a prime number
a prime power is a positive integer that is a positive integer power of a single prime number. For example: 7 = 71, 9 = 32 and 64 = 26 are prime powers
Prime_power
Infinite integer series where the next number is the sum of the two preceding it
L5466311, with 1,142,392 decimal digits. If Ln is prime then n is 0, prime, or a power of 2. L2m is prime for m = 1, 2, 3, and 4 and no other known values
Lucas_number
Number used to approximate the square root of 2
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, 44560482149, 1746860020068409
Pell_number
Technique in analytic number theory by Helmut Maier
where the difference is the primorial. By Dirichlet's theorem on arithmetic progressions the columns will contain many primes if and only if the integer
Maier's_matrix_method
Number that is the result of operation on its own digits
recreational mathematics enthusiast. A Friedman prime is a Friedman number that is also prime. The decimal Friedman primes are: 127, 347, 2503, 12101, 12107, 12109
Friedman_number
Prime number of the form (2ᵖ+1)/3
theory, a Wagstaff prime is a prime number of the form 2 p + 1 3 {\displaystyle {{2^{p}+1} \over 3}} where p is an odd prime. Wagstaff primes are named after
Wagstaff_prime
Special type of prime number
In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem
Wolstenholme_prime
Natural number
prime numbers (2, 3, 5, and 7), and thus a primorial, where it is the least common multiple of these four prime numbers. 210 is the first primorial number
210_(number)
Natural number
23 occurs twice, since adding 23 to either the fifth or eighth primorial gives a prime number (namely 2333 and 9699713). The twenty-third highly composite
23_(number)
Result on density of prime numbers
that for any integer n > 3 {\displaystyle n>3} , there exists at least one prime number p {\displaystyle p} with n < p < 2 n − 2. {\displaystyle n<p<2n-2
Bertrand's_postulate
Number of the form (n * 2^n) - 1
infinitely many Woodall primes? More unsolved problems in mathematics Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents
Woodall_number
Type of natural number in recreational number theory
the number of prime numbers which can be obtained by permuting some or all of its digits (in base 10) is larger than the number of primes obtainable in
Primeval_number
Positive integer of the form 4n + 1
A Hilbert prime is not necessarily a prime number; for example, 21 is a composite number since 21 = 3 ⋅ 7. However, 21 is a Hilbert prime since neither
Hilbert_number
Class of numbers in number theory
numbers base 2 are exactly the Mersenne numbers. A Williams prime is a Williams number that is prime. They were considered by Hugh C. Williams. It is conjectured
Williams_number
Theorem about prime numbers
constant 223,092,870 here is the product of the prime numbers up to 23, more compactly written 23# in primorial notation. On May 17, 2008, Wróblewski and Raanan
Green–Tao_theorem
Count of the possible partitions of a set
whether infinitely many Bell numbers are also prime numbers. These are called Bell primes. The first few Bell primes are: 2, 5, 877, 27644437,
Bell_number
Integer having a non-trivial divisor
positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the natural numbers that are not prime and not a unit. For example
Composite_number
Set of numbers used in the smoothsort algorithm
and also analyzed them in some detail. A Leonardo prime is a Leonardo number that is also prime. The term "Leonardo number" was coined by Dijkstra,
Leonardo_number
Algorithm for determining whether a number is prime
i\right)=1\}} and p m # {\displaystyle p_{m}\#} is the primorial – the product of the first m {\displaystyle m} primes. For example, consider p 3 # = 2 ⋅ 3 ⋅ 5 =
Primality_test
Number equal to the sum of its proper divisors
to be prime, it is necessary that p itself be prime. However, not all numbers of the form 2 p − 1 {\displaystyle 2^{p}-1} with a prime p are prime; for
Perfect_number
Number sequence 3,0,2,3,2,5,5,7,10,...
0)\\8&2P(2)+3P(1)+2P(0)&P(2)-2P(1)+P(0)\end{array}}} The first fourteen prime Perrin numbers are In 1876 the sequence and its equation were initially
Perrin_number
Analytic function in mathematics
{(p_{r-1}\#)^{k}}{J_{k}(p_{r}\#)}}\qquad k=2,3,\ldots .} Here pn# is the primorial sequence and Jk is Jordan's totient function. The function ζ can be represented
Riemann_zeta_function
Mathematical function
\log x\right).} The first Chebyshev function is the logarithm of the primorial of x, denoted x#, as we have ϑ ( x ) = ∑ p ≤ x log p = log ∏ p ≤ x
Chebyshev_function
Number with few prime factors
k-almost prime if it has k prime factors. More formally, a number n is k-almost prime if and only if Ω(n) = k, where Ω(n) is the total number of primes in the
Almost_prime
Numeral ambigram
upside down (e.g., 69, 96, 1001). A strobogrammatic prime is a strobogrammatic number that is also a prime number, i.e., a number that is only divisible by
Strobogrammatic_number
Integer of the form 3 × 2^n – 1 for non-negative n
"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, 6143, 786431, 51539607551, 824633720831
Thabit_number
Natural number
thrice, since adding 61 to either the tenth, twelfth or seventeenth primorial gives a prime number (namely 6,469,693,291; 7,420,738,134,871; and 1,922,760
61_(number)
Number of the digit form ABABAB... and A is not equal to B
Undulating numbers with odd number of digits are palindromic. They can be prime, for example 151. The undulating number ABAB...AB with n repetitions of
Undulating_number
Odd number with specific properties
be smaller. The prime Sierpiński problem asks for the value of the smallest prime Sierpiński number, and there is an ongoing "Prime Sierpiński search"
Sierpiński_number
Number of unique ways to draw non-intersecting chords in a circle
{3}{n}}\right)^{3/2}3^{n},~n\to \infty } . A Motzkin prime is a Motzkin number that is prime. Four such primes are known: 2, 127, 15511, 953467954114363 (sequence
Motzkin_number
Multiset of prime exponents in a prime factorization
divisors: 1, 2, 4, 5, 10 and 20. The smallest number of each prime signature is a product of primorials. The first few are: 1, 2, 4, 6, 8, 12, 16, 24, 30, 32
Prime_signature
Mathematical concept
Cullen primes at The Prime Pages. The Prime Glossary: Cullen number at The Prime Pages. Chris Caldwell, The Top Twenty: Generalized Cullen at The Prime Pages
Cullen_number
Natural number
term in the prime number variation of Flavius's sieve. 127 is a Fortunate number which is linked to the primorials. 127 is the 11th super-prime. The non-printable
127_(number)
Number of the form x^y + y^x
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, 8589935681, 59604644783353249
Leyland_number
Type of natural number
... (sequence A003052 in the OEIS) A self prime is a self number that is prime. The first few self primes in base 10 are 3, 5, 7, 31, 53, 97, 211, 233
Self_number
Probable prime that is composite
pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) that is not actually prime. Pseudoprimes are classified
Pseudoprime
Number n where phi(m) is greater than phi(n) for all m greater than n
Man-Kit Shiu in 1986. As they showed, every primorial is sparsely totient. If P(n) is the largest prime factor of n, then lim inf P ( n ) / log n =
Sparsely_totient_number
Type of numeral systems
proposal is the number system with successive prime numbers as radix, whose place values are primorial numbers, considered by S. S. Pillai, Richard K
Mixed_radix
Iterative algorithm on numbers
Aliquot sequences Amicable Perfect Sociable Untouchable Primorial Euclid Fortunate Other prime factor or divisor related numbers Blum Cyclic Erdős–Nicolas
Kaprekar's_routine
Centered figurate number
superstar prime is a star prime whose prime index is also a star number. The first two such numbers are 661 and 1750255921. A reverse superstar prime is a
Star_number
Natural number
increasing arithmetic progression of n primes (in this case, n = 7) that is not a primorial (a product of the first m primes). The sum of Euler's totient function
150_(number)
Number used for counting
with identity element 1; a generator set for this monoid is the set of prime numbers. In the natural numbers, addition and multiplication are compatible
Natural_number
Class of natural numbers
superabundant number is an even integer, and it is a multiple of the k-th primorial p k # . {\displaystyle p_{k}\#.} In fact, the last exponent ak is equal
Superabundant_number
each of the primes 2 i n − 1 {\displaystyle 2^{i}n-1} for 1 ≤ i ≤ k {\displaystyle 1\leq i\leq k} is a safe prime. q# denotes the primorial 2×3×5×7×..
Bi-twin_chain
Two raised to an integer power
32 × 15. A prime number that is one less than a power of two is called a Mersenne prime. For example, the prime number 31 is a Mersenne prime because it
Power_of_two
Sequence of integers
Aliquot sequences Amicable Perfect Sociable Untouchable Primorial Euclid Fortunate Other prime factor or divisor related numbers Blum Cyclic Erdős–Nicolas
Padovan_sequence
Type of number introduced by Mike Keith
Aliquot sequences Amicable Perfect Sociable Untouchable Primorial Euclid Fortunate Other prime factor or divisor related numbers Blum Cyclic Erdős–Nicolas
Keith_number
Numbers in a type of Lucas sequence
(sequence A001045 in the OEIS) A Jacobsthal prime is a Jacobsthal number that is also prime. The first Jacobsthal primes are: 3, 5, 11, 43, 683, 2731, 43691,
Jacobsthal_number
Solved prime-number problem
supplied for the primorial function, n # = ∏ p ≤ n p , {\displaystyle n\#=\prod _{p\,\leq \,n}p,} where the product is taken over all prime numbers p {\displaystyle
Proof_of_Bertrand's_postulate
Mathematical concept
k < n, the polynomial k2 − k + n produces a prime number. When k is equal to n, the value cannot be prime since n2 − n + n = n2 is divisible by n. Since
Lucky_numbers_of_Euler
Natural number
5939, 5953, 5981, 5987} 9,694,845 = Catalan number 9,699,690 = eighth primorial 9,765,625 = 31252 = 255 = 510 9,800,817 = equal to the sum of the seventh
1,000,000
Integer having only small prime factors
is an integer whose prime factors are all less than or equal to n. For example, a 7-smooth number is a number in which every prime factor is at most 7
Smooth_number
Number that has fewer digits than the number of digits in its prime factorization
{\displaystyle n} has the prime factorisation n = ∏ p prime p ∣ n p v p ( n ) {\displaystyle n=\prod _{\stackrel {p\,\mid \,n}{p{\text{ prime}}}}p^{v_{p}(n)}}
Extravagant_number
Type of sequence of prime numbers
general result known on large Cunningham chains to date. q# denotes the primorial 2 × 3 × 5 × 7 × ... × q. As of 2018[update], the longest known Cunningham
Cunningham_chain
Base-4 numeral system
However, it fares no better in the localization of prime numbers (the smallest better base being the primorial base six, senary). Quaternary shares with all
Quaternary_numeral_system
Product of two distinct primes ≡ 3 (mod 4)
Blum integer if n = p × q is a semiprime for which p and q are distinct prime numbers congruent to 3 mod 4. That is, p and q must be of the form 4t +
Blum_integer
Natural number
1-automorphic number 222,222,227 = safe prime 223,092,870 = the product of the first nine prime numbers, thus the ninth primorial 225,058,681 = Pell number 225
100,000,000
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