3/2, 4/3, 9/7, 13/10, 17/13, 47/36, 64/49, 81/62, 145/111, 226/173, 307/235, 840/643, 1147/878, 3134/2399, 4281/3277, 5428/4155, 6575/5033, 12003/9188, 221482/169539, 233485/178727, 245488/187915, 257491/197103, 269494/206291, 281497/215479, 293500/224667, 305503/233855, 317506/243043, 329509/252231, 341512/261419, 353515/270607, 365518/279795, 377521/288983, 389524/298171, 401527/307359, 413530/316547, 425533/325735, 4692866/3592273, 5118399/3918008, 5543932/4243743, 5969465/4569478, 6394998/4895213, 6820531/5220948, 7246064/5546683,7671597/5872418, 8097130/6198153, 8522663/6523888, 8948196/6849623, 9373729/7175358, 27695654/21200339, 37069383/28375697, 46443112/35551055, 148703065/113828523, 195146177/149379578, 241589289/184930633, 436735466/334310211, 1115060221/853551055, 1551795687/1187861266, 1988531153/1522171477, 3540326840/2710032743, 33414737247/25578155953, ... Generalisations There is nothing special about the middle exponent value of 3. It is possible to produce similar prime-generating functions for different middle exponent values. In fact, for any real number above 2.106..., it is possible to find a different constant A that will work with this middle exponent to always produce primes. Moreover, if Legendre's conjecture is true, the middle exponent can be replaced with value 2 . Matomäki showed unconditionally (without assuming Legendre's conjecture) the existence of a (possibly large) constant A such that is prime for all n. Additionally, Tóth proved that the floor function in the formula could be replaced with the ceiling function, so that there exists a constant such that is also prime-representing for . In the case , the value of the constant begins with 1.24055470525201424067... The first few primes generated are: Without assuming the Riemann hypothesis, Elsholtz proved that is prime for all positive integers , where , and that is prime for all positive integers , where . See also Formula for primes References Further reading External links Who remembers the Mills number?, E. Kowalski. Awesome Prime Number Constant, Numberphile. Mathematical constants Prime numbers