for integers x, y ≥ 2, where π(z) denotes the prime-counting function, giving the number of prime numbers up to and including z.
Connection to the first Hardy–Littlewood conjecture
The statement of the second Hardy–Littlewood conjecture is equivalent to the statement that the number of primes from x + 1 to x + y is always less than or equal to the number of primes from 1 to y. This was proved to be inconsistent with the first Hardy–Littlewood conjecture on prime k-tuples, and the first violation is expected to likely occur for very large values of x.[2][3] For example, an admissible k-tuple (or prime constellation) of 447 primes can be found in an interval of y = 3159 integers, while π(3159) = 446. If the first Hardy–Littlewood conjecture holds, then the first such k-tuple is expected for x greater than 1.5 × 10174 but less than 2.2 × 101198.[4]
References
^Hardy, G. H.; Littlewood, J. E. (1923). "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes". Acta Math.44 (44): 1–70. doi:10.1007/BF02403921..
^Richards, Ian (1974). "On the Incompatibility of Two Conjectures Concerning Primes". Bull. Amer. Math. Soc. 80: 419–438. doi:10.1090/S0002-9904-1974-13434-8.
second, hardy, littlewood, conjecture, number, theory, second, hardy, littlewood, conjecture, concerns, number, primes, intervals, along, with, first, hardy, littlewood, conjecture, second, hardy, littlewood, conjecture, proposed, hardy, john, edensor, littlew. In number theory the second Hardy Littlewood conjecture concerns the number of primes in intervals Along with the first Hardy Littlewood conjecture the second Hardy Littlewood conjecture was proposed by G H Hardy and John Edensor Littlewood in 1923 1 Second Hardy Littlewood conjecturePlot of p x p y p x y displaystyle pi x pi y pi x y for x y 200 displaystyle x y leq 200 FieldNumber theoryConjectured byG H HardyJohn Edensor LittlewoodConjectured in1923Open problemyes Contents 1 Statement 2 Connection to the first Hardy Littlewood conjecture 3 References 4 External linksStatement EditThe conjecture states thatp x y p x p y displaystyle pi x y leq pi x pi y for integers x y 2 where p z denotes the prime counting function giving the number of prime numbers up to and including z Connection to the first Hardy Littlewood conjecture EditThe statement of the second Hardy Littlewood conjecture is equivalent to the statement that the number of primes from x 1 to x y is always less than or equal to the number of primes from 1 to y This was proved to be inconsistent with the first Hardy Littlewood conjecture on prime k tuples and the first violation is expected to likely occur for very large values of x 2 3 For example an admissible k tuple or prime constellation of 447 primes can be found in an interval of y 3159 integers while p 3159 446 If the first Hardy Littlewood conjecture holds then the first such k tuple is expected for x greater than 1 5 10174 but less than 2 2 101198 4 References Edit Hardy G H Littlewood J E 1923 Some Problems of Partitio Numerorum III On the Expression of a Number as a Sum of Primes Acta Math 44 44 1 70 doi 10 1007 BF02403921 Hensley Douglas Richards Ian Primes in intervals Acta Arith 25 1973 74 375 391 doi 10 4064 aa 25 4 375 391 MR 0396440 Richards Ian 1974 On the Incompatibility of Two Conjectures Concerning Primes Bull Amer Math Soc 80 419 438 doi 10 1090 S0002 9904 1974 13434 8 447 tuple calculations Retrieved 2008 08 12 External links EditEngelsma Thomas J k tuple Permissible Patterns Retrieved 2008 08 12 Oliveira e Silva Tomas Admissible prime constellations Retrieved 2008 08 12 This number theory related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Second Hardy Littlewood conjecture amp oldid 1118552460, wikipedia, wiki, book, books, library,
