| Total no. of terms | Infinity |
|---|---|
| Subsequence of | Polyhedral numbers |
| Formula | |
| First terms | 1, 33, 155, 427, 909, 1661 |
| OEIS index |
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In mathematics, 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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The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if a term is even, the next term is one half of it. If a term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence. The conjecture has been shown to hold for all positive integers up to 2.95×1020, but no general proof has been found.
In mathematics, the Euler numbers are a sequence En of integers defined by the Taylor series expansion
In number theory, 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 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 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 1878 and subsequently proved by Derrick Henry Lehmer in 1930.
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.
In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum. These objects are named after Carl Friedrich Gauss, who studied them extensively and applied them to quadratic, cubic, and biquadratic reciprocity laws.
A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, or base, and they are all different, this article presents rules and examples only for decimal, or base 10, numbers. Martin Gardner explained and popularized these rules in his September 1962 "Mathematical Games" column in Scientific American.
In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by Karl von Staudt and Thomas Clausen.
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 number theory, a branch of mathematics, the Carmichael functionλ(n) of a positive integer n is the smallest positive integer m such that
The Schönhage–Strassen algorithm is an asymptotically fast multiplication algorithm for large integers, published by Arnold Schönhage and Volker Strassen in 1971. It works by recursively applying fast Fourier transform (FFT) over the integers modulo . The run-time bit complexity to multiply two n-digit numbers using the algorithm is in big O notation.
The Cunningham Project is a collaborative effort started in 1925 to factor numbers of the form bn ± 1 for b = 2, 3, 5, 6, 7, 10, 11, 12 and large n. The project is named after Allan Joseph Champneys Cunningham, who published the first version of the table together with Herbert J. Woodall. There are three printed versions of the table, the most recent published in 2002, as well as an online version by Samuel Wagstaff.
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 additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as:
In cryptography, the Rabin signature algorithm is a method of digital signature originally proposed by Michael O. Rabin in 1978.
The digital root of a natural number in a given radix is the value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached. For example, in base 10, the digital root of the number 12345 is 6 because the sum of the digits in the number is 1 + 2 + 3 + 4 + 5 = 15, then the addition process is repeated again for the resulting number 15, so that the sum of 1 + 5 equals 6, which is the digital root of that number. In base 10, this is equivalent to taking the remainder upon division by 9, which allows it to be used as a divisibility rule.
Pocklington's algorithm is a technique for solving a congruence of the form
