"this decreases exponentially with the number of tests, it provides high confidence (although not certainty) that a number that passes the repeated test is prime. On the other hand, if the test ever fails, then the number is certainly composite. A composite number that passes such a test is called a pseudoprime. In contrast, some other algorithms guarantee that their answer will always be correct: primes will always be determined to be prime and composites will always be determined to be composite. For instance, this is true of trial division. The algorithms with guaranteed-correct output include both deterministic (non-random) algorithms,"
"the smallest prime number, and the only even prime number (for this reason it is sometimes called ""the oddest prime""). The next prime is three. Two and three are the only two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, and the first Ramanujan prime. Two is the third (or fourth) Fibonacci number. Two is the base of the binary system, the numeral system with the least number of tokens allowing to denote a natural number substantially more concise ( tokens), compared to a direct representation by the corresponding count"
"also gave bounds for the smallest ""f""(""m"") needed to guarantee that infinitely many intervals of width ""f""(""m"") contain at least ""m"" primes. Usually the pair (2, 3) is not considered to be a pair of twin primes. Since 2 is the only even prime, this pair is the only pair of prime numbers that differ by one; thus twin primes are as closely spaced as possible for any other two primes. The first few twin prime pairs are: Five is the only prime in two distinct pairs. Every twin prime pair except formula_1 is of the form formula_2 for some"
"an Eisenstein prime with no imaginary part and real part of the form . It is also the only number that is part of more than one pair of twin primes. Five is a congruent number. Five is conjectured to be the only odd untouchable number and if this is the case then five will be the only odd prime number that is not the base of an aliquot tree. Five is also the only prime that is the sum of two consecutive primes, namely 2 and 3. The number 5 is the fifth Fibonacci number, being plus . It"
"leads to a different but equivalent definition of the primes: they are the numbers with exactly two positive divisors, 1 and the number itself. Yet another way to say the same thing is that a number formula_1 is prime if it is greater than one and if none of the numbers formula_10 divides formula_1 evenly. The first 25 prime numbers (all the prime numbers less than 100) are: No even number formula_1 greater than 2 is prime because any such number can be expressed as the product formula_13. Therefore, every prime number other than 2 is an odd number, and"
"Twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ""twin prime"" is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get"
"are both composite. The first few isolated primes are It follows from Brun's theorem that almost all primes are members of this sequence. Twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ""twin prime"" is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes become increasingly rare as"
"Singly and doubly even In mathematics an even integer, that is, a number that is divisible by 2, is called evenly even or doubly even if it is a multiple of 4, and oddly even or singly even if it is not. (The former names are traditional ones, derived from the ancient Greek; the latter have become common in recent decades.) These names reflect a basic concept in number theory, the 2-order of an integer: how many times the integer can be divided by 2. This is equivalent to the multiplicity of 2 in the prime factorization. A singly even"
"2 2 (two) is a number, numeral, and glyph. It is the natural number following 1 and preceding 3. An integer is called ""even"" if it is divisible by 2. For integers written in a numeral system based on an even number, such as decimal, hexadecimal, or in any other base that is even, divisibility by 2 is easily tested by merely looking at the last digit. If it is even, then the whole number is even. In particular, when written in the decimal system, all multiples of 2 will end in 0, 2, 4, 6, or 8. Two is"
"2,610,944 prime numbers ""p"" up to 43,112,609, 2 − 1 is prime for only 47 of them. Nicomachus (60–120 AD) conjectured that every perfect number is of the form 2(2 − 1) where 2 − 1 is prime. Ibn al-Haytham (Alhazen) circa 1000 AD conjectured that every ""even"" perfect number is of that form. It was not until the 18th century that Leonhard Euler proved that the formula 2(2 − 1) will yield all the even perfect numbers. Thus, there is a one-to-one correspondence between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and"
"the only prime number which is the sum of four consecutive primes (2,3,5,7). Any other four consecutive primes summed would always produce an even number, thereby divisible by 2 and so not prime. In Catalan, 17 is the first compound number (""disset""). The numbers 11 (""onze"") through 16 (""setze"") have their own names. In French, 17 is the first compound number (""dix-sept""). The numbers 11 (""onze"") through 16 (""seize"") have their own names. In Italian, 17 is also the first compound number (""diciassette""), whereas sixteen is ""sedici"". Seventeen is: 17 (number) 17 (seventeen) is the natural number following 16 and"
"""m"" + 6 is also prime then the three primes are called a prime triplet. For a twin prime pair of the form (6""n"" − 1, 6""n"" + 1) for some natural number ""n"" > 1, ""n"" must have units digit 0, 2, 3, 5, 7, or 8 (). An isolated prime (also known as single prime or non-twin prime) is a prime number ""p"" such that neither ""p"" − 2 nor ""p"" + 2 is prime. In other words, ""p"" is not part of a twin prime pair. For example, 23 is an isolated prime, since 21 and 25"
"Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: The conjecture has been shown to hold for all integers less than 4 × 10, but remains unproven despite considerable effort. A Goldbach number is a positive even integer that can be expressed as the sum of two odd primes. Since four is the only even number greater than two that requires the even prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all"
"that a regular -gon can be constructed with straightedge and compass if is a power of 2. If is a power of an odd prime number the formula for the totient says its totient can be a power of two only if is a first power and is a power of 2. The primes that are one more than a power of 2 are called Fermat primes, and only five are known: 3, 5, 17, 257, and 65537. Fermat and Gauss knew of these. Nobody has been able to prove whether there are any more. Thus, a regular -gon has"
"every even number less than ""N"" is the sum of two primes, with at most formula_15 exceptions. In particular, the set of even integers which are not the sum of two primes has density zero. Linnik proved in 1951 the existence of a constant ""K"" such that every sufficiently large even number is the sum of two primes and at most ""K"" powers of 2. Roger Heath-Brown and Jan-Christoph Schlage-Puchta in 2002 found that ""K"" = 13 works.. As with many famous conjectures in mathematics, there are a number of purported proofs of the Goldbach conjecture, none of which are"
"10). Any repunit prime is a permutable prime with the above definition, but some definitions require at least two distinct digits. All permutable primes of two or more digits are composed from the digits 1, 3, 7, 9, because no prime number except 2 is even, and no prime number besides 5 is divisible by 5. It is proven that no permutable prime exists which contains three different of the four digits 1, 3, 7, 9, as well as that there exists no permutable prime composed of two or more of each of two digits selected from 1, 3, 7,"
"The partial sum of the first pronic numbers is twice the value of the th tetrahedral number: The th pronic number is the sum of the first even integers. It follows that all pronic numbers are even, and that 2 is the only prime pronic number. It is also the only pronic number in the Fibonacci sequence and the only pronic Lucas number. The number of off-diagonal entries in a square matrix is always a pronic number. The fact that consecutive integers are coprime and that a pronic number is the product of two consecutive integers leads to a number"
"is the only triplet that can ever be prime. 111 is not prime in base ten, but is prime in base two, where 111 = 7. It is also prime in these other bases up to 128: 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119 In base 18, the number 111 is 7 (= 343) which is the only base where 111 is a perfect power. The smallest magic square using only"
"numbers are {1, 3, 5, 7, ...}. Any even number ""m"" has the form where ""k"" is again an integer. Similarly, the first non-negative even numbers are {0, 2, 4, 6, ...}. A prime number is an integer greater than 1 that is not the product of two smaller positive integers. The first few prime numbers are 2, 3, 5, 7, and 11. There is no such simple formula as for odd and even numbers to generate the prime numbers. The primes have been widely studied for more than 2000 years and have led to many questions, only some of"
"only even prime number, and 3, the first odd prime. All other prime gaps are even. There is only one pair of consecutive gaps having length 2: the gaps ""g"" and ""g"" between the primes 3, 5, and 7. For any integer ""n"", the factorial ""n""! is the product of all positive integers up to and including ""n"". Then in the sequence the first term is divisible by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of ""n"" − 1 consecutive composite integers, and it must belong to a gap between primes"
"is a unique prime of period ""n"" in base 2 if and only if there exists a natural number ""c"" such that The only known values of ""n"" such that formula_53 is composite but formula_54 is prime are 18, 20, 21, 54, 147, 342, 602, and 889 (in these case, formula_53 has a small factor which divides ""n""). It is a conjecture that there is no other ""n"" with this property. All other known base 2 unique primes are of the form formula_53. In fact, no prime with ""c"" > 1 (that is formula_54 is a ""true power"" of ""p"")"
"only odd factor on the left side, so For this equality to be true, there can be no other divisors. Therefore, ""y"" must be 1, and ""x"" must be a prime of the form Euclid–Euler theorem The Euclid–Euler theorem is a theorem in mathematics that relates perfect numbers to Mersenne primes. It states that every even perfect number has the form 2(2 − 1), where 2 − 1 is a prime number. The prime numbers of the form 2 − 1 are known as Mersenne primes, and require ""n"" itself to be prime. It has been conjectured that there are"
"are infinitely many Mersenne primes. As well as having the form 2(2 − 1), each even perfect number is the triangular number (and hence equal to the sum of the integers from 1 to ) and the hexagonal number. Furthermore, each even perfect number except for 6 is the centered nonagonal number and is equal to the sum of the first odd cubes: Even perfect numbers (except 6) are of the form with each resulting triangular number T = 28, T = 496, T = 8128 (after subtracting 1 from the perfect number and dividing the result by 9) ending"
"value of f(x) is conjectured to equal twice the twin prime constant () (not to be confused with Brun's constant), according to the Hardy–Littlewood conjecture. Every third odd number is divisible by 3, which requires that no three successive odd numbers can be prime unless one of them is 3. Five is therefore the only prime that is part of two twin prime pairs. The lower member of a pair is by definition a Chen prime. It has been proven that the pair (""m"", ""m"" + 2) is a twin prime if and only if If ""m"" − 4 or"
"68 (number) 68 (sixty-eight) is the natural number following 67 and preceding 69. It is an even number. 68 is a Perrin number. It is the largest known number to be the sum of two primes in exactly two different ways: 68 = 7 + 61 = 31 + 37. All higher even numbers that have been checked are the sum of three or more pairs of primes; the conjecture that 68 is the largest number with this property is closely related to the Goldbach conjecture and like it remains unproven. Because of the factorization of 68 as , a"
"formula_2 is a prime of the form formula_3 for prime formula_4—what is now called a Mersenne prime. Two millenia later, Euler proved that all even perfect numbers are of this form. This is known as the Euclid–Euler theorem. It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. The first perfect number is 6. Its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 +"
"for many numbers. Furthermore, all even perfect numbers besides 6 have 44 as the final two digits when expressed in senary, which is proven by the fact that every even perfect number is of the form 2(2−1), where 2−1 is prime. Senary is also the largest number base ""r"" that has no totatives other than 1 and ""r"" − 1, making its multiplication table highly regular for its size, minimizing the amount of effort required to memorize its table. This property maximizes the probability that the result of an integer multiplication will end in zero, given that neither of its"
"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 is 1 less than 32 (2). Similarly, a prime number (like 257) that is one more than a positive power of two is called a Fermat prime—the exponent itself is a power of two. A fraction that has a power of two as its denominator is called a dyadic rational. The numbers that can be represented as sums of consecutive positive integers are called polite numbers; they are exactly the numbers that are"
"239 (number) 239 (two hundred [and] thirty-nine) is the natural number following 238 and preceding 240. It is a prime number. The next is 241, with which it forms a pair of twin primes; hence, it is also a Chen prime. 239 is a Sophie Germain prime and a Newman–Shanks–Williams prime. It is an Eisenstein prime with no imaginary part and real part of the form 3""n"" − 1 (with no exponentiation implied). 239 is also a happy number. 239 is the smallest positive integer ""d"" such that the imaginary quadratic field Q() has class number = 15. HAKMEM (incidentally"
"positive even number ""k"" occurs as a prime gap infinitely often. The case ""k"" = 2 is the twin prime conjecture. The conjecture has not yet been proven or disproven for any specific value of ""k"", but Zhang Yitang's result proves that it is true for at least one (currently unknown) value of ""k"" which is smaller than 70,000,000; as discussed above, this has been improved to 246. The gap ""g"" between the ""n""th and (""n"" + 1)st prime numbers is an example of an arithmetic function. In this context it is usually denoted ""d"" and called the prime difference"
"Chen prime A prime number ""p"" is called a Chen prime if ""p"" + 2 is either a prime or a product of two primes (also called a semiprime). The even number 2""p"" + 2 therefore satisfies Chen's theorem. The Chen primes are named after Chen Jingrun, who proved in 1966 that there are infinitely many such primes. This result would also follow from the truth of the twin prime conjecture. The first few Chen primes are The first few Chen primes that are not the lower member of a pair of twin primes are The first few non-Chen primes"
"the unique prime that has a finite reciprocal in base . For example, 2 is not considered as a unique prime in binary, although it is the only prime with finite reciprocal in binary. The mention ""terminated"" means that the prime divides the base, and thus that the representation of its reciprocal is finite. Bold for unique primes. At present, more than fifty unique primes or probable primes are known. However, there are only twenty-three unique primes below 10. The following table lists all 23 unique primes below 10 (sequence (sorted) and (ordered by period length) in OEIS) and their"
"odd primes are 26861 and 26862. The sum of one odd square and one even square is congruent to 1 mod 4, but there exist composite numbers such as 21 that are 1 mod 4 and yet cannot be represented as sums of two squares. Fermat's theorem on sums of two squares states that the prime numbers that can be represented as sums of two squares are exactly 2 and the odd primes congruent to 1 mod 4. The representation of each such number is unique, up to the ordering of the two squares. By using the Pythagorean theorem, this"