| 词条 | Second Hardy–Littlewood conjecture |
| 释义 |
In number theory, the second Hardy–Littlewood conjecture concerns the number of primes in intervals. The conjecture states that π(x + y) ≤ π(x) + π(y) for x, y ≥ 2, where π(x) denotes the prime-counting function, giving the number of prime numbers up to and including x. This means 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.[1][2] For example, an admissible k-tuple [3] (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] References1. ^{{cite journal | first=Douglas | last=Hensley | first2=Ian | last2=Richards | title=Primes in intervals | journal=Acta Arith. | volume=25 | issue=1973/74 | pages=375–391 | mr=396440 }} * {{cite web | first=Thomas J. | last=Engelsma | title=k-tuple Permissible Patterns | url=http://www.opertech.com/primes/k-tuples.html | accessdate=2008-08-12 }}2. ^{{cite journal | first=Ian | last=Richards | title=On the Incompatibility of Two Conjectures Concerning Primes | journal=Bull. Amer. Math. Soc. | volume=80 | pages=419–438 | year=1974 | doi=10.1090/S0002-9904-1974-13434-8 }} 3. ^{{cite web | title=Prime pages: k-tuple | url=http://primes.utm.edu/glossary/page.php?sort=ktuple | accessdate=2008-08-12}} 4. ^{{cite web | title=447-tuple calculations | url=http://www.opertech.com/primes/residues.html | accessdate=2008-08-12}}
2 : Analytic number theory|Conjectures about prime numbers |
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