Twin Primes Calculator (Pairs, Brun, Hardy-Littlewood)

Twin primes calculator: check (p, p+2) pairs, enumerate up to N, Brun constant B_2 and Hardy-Littlewood asymptotic with C_2 = 0.6602 twin prime constant.

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Twin primes

Two primes p and q are twin if q = p + 2 (smallest possible difference between odd primes). First examples: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151). The twin prime conjecture, one of number theory's most famous open problems, states infinitely many such pairs exist. Formulated by Alphonse de Polignac in 1849, resists for nearly two centuries.

Fundamental properties

Except (3, 5), all twin pairs have form (6k-1, 6k+1). Proof: any odd prime p > 3 is congruent to 1 or 5 mod 6 (else divisible by 2 or 3). For p+2 prime, p = 6k-1 -> p+2 = 6k+1, both in {-1, 1} mod 6. Except (3, 5), every twin pair has center 6k divisible by 6.

Triplets and rarity

A triplet would be (p, p+2, p+4) all prime. Only one exists: (3, 5, 7). For p > 3, one of {p, p+2, p+4} is always divisible by 3 (mod 3 covers {0, 1, 2}). So (3, 5, 7) is the unique triplet.

Brun's constant

Viggo Brun (1919) proved a surprising result: the sum of twin prime reciprocals converges:

B_2 = (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17 + 1/19) + ... = 1.902160583104...

Unlike Sigma 1/p on all primes (diverges as log log n), the twin series converges. First step toward understanding twin rarity. Exact value of B_2 is still unknown: exact computation requires enumeration of twins up to astronomical limits.

Hardy-Littlewood conjecture (1923)

Hardy and Littlewood gave a quantitative estimate of twin pair count up to N:

pi_2(N) ~ 2 * C_2 * N / log^2(N)

where C_2 = 0.6601618158... is the twin prime constant, defined as product:

C_2 = Prod_{p prime, p >= 3} [1 - 1/(p-1)^2]

Empirical agreement excellent: for N=10^10 prediction is ~20.77 million pairs vs 27.4 million actual (deviation <0.1%). Doesn't imply infinite twins but quantifies asymptotic frequency assuming conjecture.

Current research status

  • Zhang (2013): infinitely many prime pairs (p, q) with q - p < 70'000'000, reduced to 246 (Polymath8 2014) and 12 under GRH (Maynard 2015).
  • Bound below 6 requires still-open hypotheses (Elliott-Halberstam).
  • Largest known twin pair: 2996863034895 * 2^1290000 +/- 1 (409'000 digits), verified 2016 by PrimeGrid.
  • Related families: sexy primes (p, p+6), cousin primes (p, p+4), Chen primes.
  • Chen's theorem (1966): infinitely many primes p with p+2 at most product of two primes.

Five calculator modes

  1. Check: given n, checks if (n, n+2) or (n-2, n) is twin pair.
  2. Find near: all twin pairs in [n-500, n+500].
  3. List up to N: full enumeration via Eratosthenes sieve.
  4. Brun constant: partial estimate of B_2 summing reciprocals.
  5. Hardy-Littlewood: compare empirical pi_2(N) vs theoretical prediction.

Sexy, cousin, triple primes

Beyond twins (gap 2), other similar families exist:

  • Cousin primes: (p, p+4). Examples: (3, 7), (7, 11), (13, 17), (19, 23).
  • Sexy primes: (p, p+6). Examples: (5, 11), (7, 13), (11, 17), (13, 19).
  • Prime triplets: (p, p+2, p+6) or (p, p+4, p+6). Only (3, 5, 7) is type 2,4.
  • Prime quadruplets: (p, p+2, p+6, p+8). Examples: (5, 7, 11, 13), (11, 13, 17, 19).
  • All families have analogous Brun constants and density conjectures.

Applications

  • Analytic number theory: Hardy-Littlewood circle method.
  • Cryptography: RSA generation uses 'strong' primes sometimes selected among twins.
  • Distributed computing: PrimeGrid, Twin Prime Search projects.
  • Sieve testing: performance of prime sieve algorithms (Eratosthenes, Atkin, segmented).
  • Sieve theory: Selberg, Chen, Brun techniques applied beyond twins.
  • Education: accessible example of modern open problem.

From Euclid to Zhang: the long hunt for twin primes

The question of whether twin primes are infinite is among the oldest and most stubborn in mathematics. Euclid already proved in the 3rd century BC that primes are infinite, but the twin question remained open for over two millennia. A breakthrough came in 2013, when the unknown mathematician Yitang Zhang proved that infinitely many prime pairs exist with a distance below 70 million: the first time a finite bound was established. The collaborative Polymath project, coordinated by Terence Tao, quickly reduced that bound to 246. Reducing it to 2, proving the original conjecture, remains one of the great open problems. A curious twin-prime episode was Thomas Nicely's 1994 discovery of a bug in Intel's Pentium processor while computing Brun's constant, an error that cost the company hundreds of millions of dollars.

Updated July 2026

Frequently Asked Questions

Are there infinitely many twin primes?

Conjectured but unproven. Very strong empirical evidence: up to 10^18 about 808 trillion twin pairs, their count follows Hardy-Littlewood prediction accurately. Zhang (2013) proved infinite prime pairs with difference < 70 million, reduced to 246 (Maynard-Tao 2014) and 12 under GRH. Gap 2 remains open.

What exactly does Zhang 2013 say?

Yitang Zhang proved (2013) existence of infinitely many n with n and n+k both prime for some k <= 70'000'000. First finite 'bounded gaps' result - bound exists. Terry Tao's Polymath8 Project collaborative reduced bound to 4680 then 246 (Maynard). With Elliott-Halberstam, bound <= 6; but 2 (twin conjecture) remains open. Zhang awarded MacArthur Grant and Cole Prize.

Why is Brun convergent surprising?

Sigma 1/p on all primes diverges (Euler 1737) as log log n, showing primes 'dense enough'. But Sigma (1/p + 1/q) on twins converges (Brun 1919). Implies twins become rare quickly: density ~ C/log^2(n), integrable. Historically, Brun's convergence destroyed hopes of 'easy' infinity proof via Euler methods.

Largest known twin pair?

2996863034895 * 2^1290000 +/- 1, 409'000 digits, discovered 2016 by PrimeGrid. Calculation required thousands of CPU-hours and Lucas-Lehmer-Riesel test. Smaller but notable: 65516468355 * 2^333333 +/- 1 (100'000 digits), 3756801695685 * 2^666669 +/- 1 (200'000 digits). Record grows regularly via distributed computing.

Why is (3, 5, 7) the only triplet?

For p > 3, among three consecutive numbers p, p+2, p+4 one is divisible by 3 (mod 3 covers {0, 1, 2}). If p = 3, (3, 5, 7) works. For p > 3 the multiple of 3 isn't prime (not 3). So no other triplet. Similarly (p, p+2, p+6) and (p, p+4, p+6) can exist for infinitely many p (conjectured) but not (p, p+2, p+4) for p > 3.

How to compute Brun's constant?

Requires twin enumeration up to large limits and summing reciprocals. B_2 = Sigma(1/p + 1/q). Convergence very slow: value at N=10^13 is ~1.833, far from 1.902. Richardson extrapolation and Hardy-Littlewood asymptotic used to reduce error. Kutsia et al. (2002) computed B_2 = 1.902160583104... with ~10 verified digits. Exact value unknown.

What is Chen's theorem (1966)?

Chen Jingrun proved a result considered 'almost' Goldbach and twin conjecture: infinitely many primes p with p+2 prime or semiprime (product of two primes). Proof uses weighted Chen sieve, modification of Brun sieve. Strongest result 'toward' twin primes, but distinguishing 'prime' from 'semiprime' in Chen's conclusion requires still-undeveloped techniques.

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