| Total no. of terms | Infinity |
|---|---|
| Subsequence of | Polyhedral numbers |
| Formula | |
| First terms | 1, 33, 155, 427, 909, 1661 |
| OEIS index |
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A centered dodecahedral number is a centered figurate number that represents a dodecahedron. The centered dodecahedral number for a specific n is given by
The first such numbers are 1, 33, 155, 427, 909, 1661, 2743, 4215, 6137, 8569, … (sequence A005904 in the OEIS ).
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In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number p: its value at a (nonzero) quadratic residue mod p is 1 and at a non-quadratic residue (non-residue) is −1. Its value at zero is 0.
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.
Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as
In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific conditions. While there may be probable primes that are composite, the condition is generally chosen in order to make such exceptions rare.
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function is a Dirichlet character of modulus if for all integers and :
In modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which gk ≡ a. Such a value k is called the index or discrete logarithm of a to the base g modulo n. So g is a primitive root modulo n if and only if g is a generator of the multiplicative group of integers modulo n.
In theoretical computer science and cryptography, a trapdoor function is a function that is easy to compute in one direction, yet difficult to compute in the opposite direction without special information, called the "trapdoor". Trapdoor functions are a special case of one-way functions and are widely used in public-key cryptography.
In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n. That is, the factorial satisfies
In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known.
In mathematics, the Lucas–Lehmer test (LLT) is a primality test for Mersenne numbers. The test was originally developed by Édouard Lucas in 1876 and subsequently improved by Derrick Henry Lehmer in the 1930s.
In number theory, a congruence of squares is a congruence commonly used in integer factorization algorithms.
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known. It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve. It is a general-purpose factorization algorithm, meaning that its running time depends solely on the size of the integer to be factored, and not on special structure or properties. It was invented by Carl Pomerance in 1981 as an improvement to Schroeppel's linear sieve.
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie-Hellman Key Exchange and RSA public/private keys.
A strong pseudoprime is a composite number that passes the Miller–Rabin primality test. All prime numbers pass this test, but a small fraction of composites also pass, making them "pseudoprimes".
In classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. Invented by Lester S. Hill in 1929, it was the first polygraphic cipher in which it was practical to operate on more than three symbols at once.
In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number p, then this root can be lifted to a unique root modulo any higher power of p. More generally, if a polynomial factors modulo p into two coprime polynomials, this factorization can be lifted to a factorization modulo any higher power of p.
In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. In the standard notation of modular arithmetic this congruence is written as
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x4 ≡ p is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence x4 ≡ p to that of x4 ≡ q.
In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined as
Pocklington's algorithm is a technique for solving a congruence of the form
