Search for a counterexample to $\pi(x+y)\le\pi(x)+\pi(y)$ (second Hardy–Littlewood conjecture, Erdős #855)
Statement
Let $\pi(t)$ count the primes in $[1,t]$. The second Hardy–Littlewood conjecture asserts $\pi(x+y)\le\pi(x)+\pi(y)$ for all integers $x,y\ge 2$ — equivalently, no interval of length $y$ contains more primes than the initial interval $[2,y+1]$. Is this true, or does a counterexample exist?
Acceptance. FULLY RESOLVES (disproof): exhibit explicit integers $x,y$ with $\pi(x+y)>\pi(x)+\pi(y)$ — machine-verifiable by two prime counts. PARTIAL: (a) verify $\pi(x+y)\le\pi(x)+\pi(y)$ for all $x,y$ up to a stated bound; or (b) construct an admissible tuple of length $y$ containing more than $\pi(y)$ elements (a counterexample template, unconditional on primality). Provide code and certificate.
Background
Erdős Problem #855 (Erdős raised it repeatedly: Er61, Er65b, Er80 p.108, Er82e, Er85c). Hensley & Richards (1974) proved this conjecture is **incompatible** with the prime $k$-tuples conjecture (very widely believed), so a counterexample is expected to exist, yet none has ever been exhibited and the length at which one first appears is enormous and unknown. Extremal function: $\rho(y)=\max_x(\pi(x+y)-\pi(x))$; a counterexample is a $y$ with $\rho(y)>\pi(y)$. Entry: erdosproblems.com/855.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #855 (erdosproblems.com) | link |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.