There exist a variety of formulas for either producing the th prime as a function of or taking on only prime values. However, all such formulas require either extremely accurate knowledge of some unknown constant, or effectively require knowledge of the primes ahead of time in order to use the formula (Dudley 1969; Ribenboim 1996, p. 186). There also exist simple prime-generating polynomials that generate only primes for the first (possibly large) number of integer values.There are also many beautiful formulas involving prime sums and prime products that can be done in closed form.Considering examples of formulas that produce only prime numbers (although not necessarily the complete set of prime numbers ), there exists a constant (OEIS A051021) known as Mills' constant such that(1)where is the floor function, is prime for all (Ribenboim 1996, p. 186). The first few values of are 2, 11, 1361, 2521008887, ... (OEIS A051254). It..
The prime distance of a nonnegative integer is the absolute difference between and the nearest prime. It is therefore true that for primes . The first few values for , 1, 2, ... are therefore 2, 1, 0, 0, 1, 0, 1, 0, 1, 2, ... (OEIS A051699). The values of having prime distances of 0, 1, 2, 3, ... are 2, 1, 0, 26, 93, 118, 119, 120, 531, 532, 897, ... (OEIS A077019).
For an odd prime and a positive integer which is not a multiple of ,where is the Legendre symbol.
The quotientwhich must be congruent to 0 (mod ) for to be a Wilson prime. The quotient is an integer only when (in which case ) or is a prime, and the values of corresponding to , 3, 5, 7, 11, ... are 1, 1, 5, 103, 329891, 36846277, 1230752346353, ... (OEIS A007619).
Mann's theorem is a theorem widely circulated as the " conjecture" that was subsequently proven by Mann (1942). It states that if and are sets of integers each containing 0, thenHere, denotes the direct sum, i.e., , and is the Schnirelmann density.Mann's theorem is optimal in the sense that satisfies .Mann's theorem implies Schnirelmann's theorem as follows. Let , then Mann's theorem proves that , so as more and more copies of the primes are included, the Schnirelmann density increases at least linearly, and so reaches 1 with at most copies of the primes. Since the only sets with Schnirelmann density 1 are the sets containing all positive integers, Schnirelmann's theorem follows.
Let be an irregular prime, and let be a prime with . Also let be an integer such that (mod ). For an irregular pair , form the product(1)where(2)(3)If (mod ) for all such irregular pairs, then Fermat's last theorem holds for exponent .
Chebyshev noticed that the remainder upon dividing the primes by 4 gives 3 more often than 1, as plotted above in the left figure. Similarly, dividing the primes by 3 gives 2 more often than 1 (right figure). This is called the Chebyshev bias, or sometimes the prime race (Wagon 1994).Consider the list of the first primes (mod 4). This list contains equal numbers of remainders 3 and 1 (mod 4) for , 3, 7, 13, 89, 2943, 2945, 2947, 2949, 2951, 2953, 50371, ... (OEIS A038691; Wagon 1994, pp. 2-3). The values of for which the list is biased towards 1 are 2946, 50378, 50380, 50382, 50383, 50384, 50385, ... (OEIS A096628).Definingthe values of for which are , 3, 7, 13, 89, 2943, 2945, 2947, ... (OEIS A038691).Similarly, consider the list of the first primes (mod 3), skipping and since . This list contains equal numbers of remainders 2 and 1 at the values , 6, 8, 12, 14, 22, 38, 48, 50, ... (OEIS A096629). The first value of for which the list is biased towards 1 is , as..
If is the th prime such that is a Mersenne prime, thenIt was modified by Wagstaff (1983) to yield Wagstaff'sconjecture,where is the Euler-Mascheroni constant.
Lehmer's totient problem asks if there exist any composite numbers such that , where is the totient function? No such numbers are known. However, any such an would need to be a Carmichael number, since for every element in the integers (mod ), , so and is a Carmichael number.In 1932, Lehmer showed that such an must be odd and squarefree, and that the number of distinct prime factors must satisfy . This was subsequently extended to . The best current result is and , improving the lower bound of Cohen and Hagis (1980) since there are no Carmichael numbers less than having distinct prime factors; Pinch). However, even better results are known in the special cases , in which case (Wall 1980), and , in which case and (Lieuwens 1970).
The Cramér conjecture is the unproven conjecturethatwhere is the th prime.
There exist infinitely many with for all , where is the th prime. Also, there exist infinitely many such that for all .
Andrica's conjecture states that, for the th prime number, the inequalityholds, where the discrete function is plotted above. The high-water marks for occur for , 2, and 4, with , with no larger value among the first primes. Since the Andrica function falls asymptotically as increases, a prime gap of ever increasing size is needed to make the difference large as becomes large. It therefore seems highly likely the conjecture is true, although this has not yet been proven. bears a strong resemblance to the prime difference function, plotted above, the first few values of which are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, ... (OEIS A001223).A generalization of Andrica's conjecture considers the equationand solves for . The smallest such is (OEIS A038458), known as the Smarandache constant, which occurs for and (Perez)...
Let be the th Bernoulli number and considerwhere the residues of fractions are taken in the usual way so as to yield integers, for which the minimal residue is taken. Agoh's conjecture states that this quantity is iff is prime. There are no counterexamples less than (S. Plouffe, pers. comm., Jan. 28, 2003). Any counterexample to Agoh's conjecture would be a contradiction to Giuga's conjecture, and vice versa.For , 2, ..., the minimal integer residues (mod ) is 0, , , 0, , 0, , 0, , 0, , ... (OEIS A046094).Kellner (2002) provided a short proof of the equivalence of Giuga'sand Agoh's conjectures. The combined conjecture can be described by a sum of fractions.
The abc conjecture is a conjecture due to Oesterlé and Masser in 1985. It states that, for any infinitesimal , there exists a constant such that for any three relatively prime integers , , satisfying(1)the inequality(2)holds, where indicates that the product is over primes which divide the product . If this conjecture were true, it would imply Fermat's last theorem for sufficiently large powers (Goldfeld 1996). This is related to the fact that the abc conjecture implies that there are at least non-Wieferich primes for some constant (Silverman 1988, Vardi 1991).The conjecture can also be stated by defining the height and radical of the sum as(3)(4)where runs over all prime divisors of , , and . Then the abc conjecture states that for all , there exists a constant such that for all ,(5)(van Frankenhuysen 2000). van Frankenhuysen (2000) has shown that there exists an infinite sequence of sums or rational integers with large height compared..
Grimm conjectured that if , , ..., are all composite numbers, then there are distinct primes such that for .
If andis necessarily a prime? In other words, definingdoes there exist a composite such that ? It is known that iff for each prime divisor of , and (Giuga 1950, Borwein et al. 1996); therefore, any counterexample must be squarefree. A composite integer satisfies iff it is both a Carmichael number and a Giuga number. Giuga showed that there are no exceptions to the conjecture up to . This was later improved to (Bedocchi 1985) and (Borwein et al. 1996).Kellner (2002) provided a short proof of the equivalence of Giuga's and Agoh'sconjectures. The combined conjecture can be described by a sum of fractions.
A modification of the Eberhart's conjecture proposed by Wagstaff (1983) which proposes that if is the th prime such that is a Mersenne prime, thenwhere is the Euler-Mascheroni constant.
Let denote the number of primes which are congruent to modulo (i.e., the modular prime counting function). Then one might expect that(Berndt 1994).Although this is true for small numbers, Hardy and Littlewood showed that changes sign infinitely often. The effect was first noted by Chebyshev in 1853, and is sometimes called the Chebyshev phenomenon. It was subsequently studied by Shanks (1959), Hudson (1980), and Bays and Hudson (1977, 1978, 1979). The effect was also noted by Ramanujan, who incorrectly claimed that (Berndt 1994).The bias of the sign of is known as the Chebyshev bias.
Let the difference of successive primes be defined by , and by(1)N. L. Gilbreath claimed that for all (Guy 1994). In 1959, the claim was verified for . In 1993, Odlyzko extended the claim to all primes up to .Gilbreath's conjecture is equivalent to the statement that, in the triangular array of the primes, iteratively taking the absolute difference of each pair of terms(2)(OEIS A036262), always gives leading term 1(after the first row).The number of terms before reaching the first greater than two in the second, third,etc., rows are given by 3, 8, 14, 14, 25, 23, 22, 25, ... (OEIS A000232).
is prime iff the 14 Diophantine equations in 26 variables(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)have a solution in positive integers (Jones etal. 1976; Riesel 1994, p. 40).
Let be an odd prime and a positive integer not divisible by . Then for each positive odd integer , let bewith , and let be the number of even s. Thenwhere is the Legendre symbol.
The Fermat quotient for a number and a prime base is defined as(1)If , then(2)(3)(mod ), where the modulus is taken as a fractional congruence.The special case is given by(4)(5)(6)(7)(8)all again (mod ) where the modulus is taken as a fractional congruence, is the digamma function, and the last two equations hold for odd primes only. is an integer for a prime, with the values for , 3, 5, ... being 1, 3, 2, 5, 3, 13, 3, 17, 1, 6, ....The quantity is known to be congruent to zero (mod ) for only two primes: the so-called Wieferich primes 1093 and 3511 (Lehmer 1981, Crandall 1986).
If is a prime number and is a natural number, then(1)Furthermore, if ( does not divide ), then there exists some smallest exponent such that(2)and divides . Hence,(3)The theorem is sometimes also simply known as "Fermat'stheorem" (Hardy and Wright 1979, p. 63).This is a generalization of the Chinese hypothesis and a special case of Euler's totient theorem. It is sometimes called Fermat's primality test and is a necessary but not sufficient test for primality. Although it was presumably proved (but suppressed) by Fermat, the first proof was published by Euler in 1749. It is unclear when the term "Fermat's little theorem" was first used to describe the theorem, but it was used in a German textbook by Hensel (1913) and appears in Mac Lane (1940) and Kaplansky (1945).The theorem is easily proved using mathematical induction on . Suppose (i.e., divides ). Then examine(4)From the binomial theorem,(5)Rewriting,(6)But..
A prime number (or prime integer, often simply called a "prime" for short) is a positive integer that has no positive integer divisors other than 1 and itself. More concisely, a prime number is a positive integer having exactly one positive divisor other than 1, meaning it is a number that cannot be factored. For example, the only divisors of 13 are 1 and 13, making 13 a prime number, while the number 24 has divisors 1, 2, 3, 4, 6, 8, 12, and 24 (corresponding to the factorization ), making 24 not a prime number. Positive integers other than 1 which are not prime are called composite numbers.While the term "prime number" commonly refers to prime positive integers, other types of primes are also defined, such as the Gaussian primes.The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p. 31). Although the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909, 1914; Hardy and Wright..
Landau (1911) proved that for any fixed ,as , where the sum runs over the nontrivial Riemann zeta function zeros and is the Mangoldt function. Here, "fixed " means that the constant implicit in depends on and, in particular, as approaches a prime or a prime power, the constant becomes large.Landau's formula is roughly the derivative of the explicitformula.Landau's formula is quite extraordinary. If is not a prime or a prime power, then and the sum grows as a constant times . But if is a prime or a prime power, then and the sum grows much faster, like a constant times . This exhibits an amazing connection between the primes and the s; somehow the zeros "recognize" when is a prime and cause large contributions to the sum.
A number satisfies the Carmichael condition iff for all prime divisors of . This is equivalent to the condition for all prime divisors of .