For every pair of linear polynomials f_1(n)=a_1n+b_1 and f_2(n)=a_2n+b_2 with a_1,a_2>0 that is admissible—meaning that, for every prime p, there exists an integer n such that p nmid f_1(n)f_2(n)—there exist infinitely many positive integers n for which both f_1(n) and f_2(n) are prime.

Dickson’s conjecture for two linear forms

Dickson’s conjecture predicts that every admissible pair of linear forms is simultaneously prime for infinitely many integer inputs. The two-form case remains unproved.

Known results

Maynard–Tao methods prove bounded-gap and other weak prime-tuple results, but not simultaneous primality for every form in an admissible system.

September 2026 graph bounds

Yi Hu and Nan Yang prove unconditional bounds $6\leq\operatorname{diameter}\leq7$ and $4\leq\operatorname{radius}\leq5$ for an additive–multiplicative graph. Assuming Dickson’s conjecture, they obtain exact values $\operatorname{diameter}=6$ and $\operatorname{radius}=4$ ; this is progress on related consequences, not a proof of the conjecture.

Current status (as of October 2026): Dickson’s conjecture for two linear forms remains open; the latest paper supplies narrow unconditional graph bounds and conditional exact values.

Dickson’s conjecture for two linear forms

Primary source

Additional references

Progress summary

Solutions 0