Mills' constant
In number theory, Mills' constant is defined as the smallest positive real number "A" such that the floor function of the double exponential function is a prime number for all natural numbers "n". This constant is named after William H. Mills who proved in 1947 the existence of "A" based on results of Guido Hoheisel and Albert Ingham on the prime gaps. Its value is unknown, but if the Riemann hypothesis is true, it is approximately 1.3063778838630806904686144926... . Mills primes. The primes generated by Mills' constant are known as Mills primes; if the Riemann hypothesis is true, the sequence  begins If "a" denotes the "i" prime in this sequence, then "a" can be calculated as the smallest prime number larger than formula_3. In order to ensure that rounding formula_4, for "n" = 1, 2, 3, …, produces this sequence of primes, it must be the case that formula_5. The Hoheisel–Ingham results guarantee that there exists a prime between any two sufficiently large cube numbers, which is sufficient to prove this inequality if we start from a sufficiently large first prime formula_6. The Riemann hypothesis implies that there exists a prime between any two consecutive cubes, allowing the "sufficiently large  condition to be removed, and allowing the sequence of Mills primes to begin at a" = 2. For all a > formula_7, there is at least one prime between formula_8 and formula_9. This upper bound is much too large to be practical, as it is infeasible to check every number below that figure. However, the value of Mills' constant can be verified by calculating the first prime in the sequence that is greater than that figure. As of April 2017, the 11th number in the sequence is the largest one that has been "proved" prime. It is and has 20562  digits. , the largest known Mills "probable" prime (under the Riemann hypothesis) is , which is 555,154 digits long. Numerical calculation. By calculating the sequence of Mills primes, one can approximate Mills' constant as Caldwell and Cheng used this method to compute 6850 base 10 digits of Mills' constant under the assumption that the Riemann hypothesis is true. There is no closed-form formula known for Mills' constant, and it is not even known whether this number is rational. If it is rational, and if we can calculate its decimal expansion to the point where it repeats, this will allow us to  generate infinitely many provable primes. Fractional representations. Below are fractions which approximate Mills' constant, listed in order of increasing accuracy (with continued-fraction convergents in bold) : 1/1, 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 formula_13 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 formula_14 such that is also prime-representing for formula_16. In the case formula_17, the value of the constant formula_14 begins with 1.24055470525201424067... The first few primes generated are: "Without" assuming the Riemann hypothesis, Elsholtz proved that formula_20 is prime for all positive integers , where formula_21, and that formula_22 is prime for all positive integers , where formula_23.