second Hardy–Littlewood conjecture

{{Infobox mathematical statement

| name = Second Hardy–Littlewood conjecture

| image = File:Second Hardy–Littlewood conjecture.svg

| caption = Plot of $\pi(x) + \pi(y) - \pi(x+y)$ for $x, y \leq 200$

| field = Number theory

| conjectured by = G. H. Hardy
John Edensor Littlewood

| conjecture date = 1923

| open problem = yes

| first proof by =

| first proof date =

| implied by =

}}

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.{{cite journal |last1=Hardy |first1=G. H. |author-link1=G. H. Hardy |last2=Littlewood |first2=J. E. |author-link2=John Edensor Littlewood |title=Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes. |journal=Acta Math. |issue=44 |pages=1–70 |year=1923 |volume=44|doi=10.1007/BF02403921|doi-access=free }}.

Statement

The conjecture states that

$\pi(x+y) \leq \pi(x) + \pi(y)$

for integers {{Math|x, y ≥ 2}}, where {{Math|π(z)}} denotes the prime-counting function, giving the number of prime numbers up to and including {{Mvar|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 {{Math|x + 1}} to {{Math|x + y}} is always less than or equal to the number of primes from 1 to {{Mvar|y}}. This was proved to be inconsistent with the first Hardy–Littlewood conjecture on prime {{Mvar|k}}-tuples, and the first violation is expected to likely occur for very large values of {{Mvar|x}}.{{cite journal | first1=Douglas | last1=Hensley | first2=Ian | last2=Richards | title=Primes in intervals | journal=Acta Arith. | date=1974 | volume=25 | issue=1973/74 | pages=375–391 | mr=396440 | doi=10.4064/aa-25-4-375-391 | doi-access=free }}{{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 | doi-access=free }} For example, an admissible k-tuple (or prime constellation) of 447 primes can be found in an interval of {{Math|1=y = 3159}} integers, while {{Math|π(3159) {{=}} 446}}. If the first Hardy–Littlewood conjecture holds, then the first such {{Mvar|k}}-tuple is expected for {{Mvar|x}} greater than {{Math|1.5 × 10¹⁷⁴}} but less than {{Math|2.2 × 10¹¹⁹⁸}}.{{cite web | title=447-tuple calculations | url=http://www.opertech.com/primes/residues.html | accessdate=2008-08-12}}

References

External links

{{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 }}
{{cite web | first=Tomás | last=Oliveira e Silva | title=Admissible prime constellations | url=https://sweet.ua.pt/tos/apc.html | accessdate=2023-09-28 }}

Category:Analytic number theory

Category:Conjectures about prime numbers

Category:Unsolved problems in number theory