Cousin Primes

Cousin primes are pairs of primes of the form (, ). The first few are (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), ... (OEIS A023200 and A046132).

A large pair of proven cousin primes is , where

(1)

where is a primorial. These primes have 10154 digits and were found by T. Alm, M. Fleuren, and J. K. Andersen (Andersen 2005).

As of Oct. 2026, the largest known pair of cousin primes is , where

(2)

Both primes have 100,032 decimal digits and were found by S. Batalov in April 2025 (Prime -Tuplets).

Let count cousin-prime pairs whose smaller prime number does not exceed . The first Hardy-Littlewood conjecture predicts the same asymptotic form as for the twin primes,

			(3)
			(4)

where the product is over primes , is the twin primes constant, and (OEIS A114907).

The predicted asymptotic for also follows from a sufficiently strong ratios conjecture for the Riemann zeta function. Lee (2026) proved this conditional implication under a uniform two-over-two zeta-ratios formula with a square-root-size error term, with four shifts whose real parts lie in and whose imaginary parts have absolute value at most at height . Thus the conclusion remains conditional on this uniform ratios hypothesis; it is not an unconditional proof that infinitely many cousin primes exist.

By analogy with Brun's constant, the constant

(5)

(omitting the initial term ) can be defined. Using cousin primes up to and extrapolating the tail, Wolf (1996, p. 7) estimated

(6)