FANDOM


The formula n2 + n + 35 is prime for all 0≤n≤33 (this formula can generate 34 different primes for this 34 consecutive n, and can generate 68 primes for 68 consecutive n (from −34 to 33)), n2 + n + p is also prime for 0≤np−2 (since it is divisible by p for n = p−1, one cannot do be better than this) for p = 2, 3, 5, E and 15, they are called Euler's lucky numbers, and corresponding to the Heegner numbers (squarefree numbers d such that the imaginary quadratic field

$ Q[\sqrt{-d}] $ has class number 1) 7, E, 17, 37, 57 and 117 (if and only if n2 + n + p is prime for 0≤np−2, then 4p−1 is a Heegner number), there are no such p > 35, and there are no Heegner number > 117.

The formula 2n2 + 25 is prime for all 0≤n≤24 (this formula can generate 25 different primes for this 25 consecutive n, and can generate 49 primes for 49 consecutive n (from −24 to 24)), 2n2 + p is also prime for 0≤np−1 (since it is divisible by p for n = p, one cannot do be better than this) for p = 3, 5 and E, they are called Euler's quasi-lucky numbers, and corresponding to the quasi-Heegner numbers (even squarefree numbers d such that the imaginary quadratic field

$ Q[\sqrt{-d}] $ has class number 2) 6, X, 1X and 4X (if and only if 2n2 + p is prime for 0≤np−1, then 2p is a quasi-Heegner number), there are no such p > 25, and there are no quasi-Heegner number > 4X.

f(n) prime with n range number of such consecutive n number of distinct primes largest prime smallest composite
n2 + n + 35 [−34, 33] 68 34 E15 E81 = 352
n2 + n + 15 [−14, 13] 28 14 195 201 = 152
n2 + n + E [−X, 9] 18 X 85 X1 = E2
2n2 + 25 [−24, 24] 49 25 E11 EX7 = 25 × 4E
2n2 + E [−X, X] 19 E 157 191 = E × 1E
4n2 + 117 [−17, 17] 33 18 E1E 102E = 35 × 37
4n2 + 57 [−7, 7] 13 8 19E 22E = 15 × 17
4n2 + 31 [−8, 8] 15 9 205 261 = 172
2n2 + 2n + 17 [−16, 15] 30 16 447 4X7 = 17 × 31
3n2 + 3n + 1E [−1X, 19] 38 1X 995 X85 = 1E × 57
4n2 + 4n + 465 [−16, 15] 30 16 1105 1205 = 37 × 3E
6n2 + 6n + 27 [−25, 24] 4X 25 2X07 3057 = 4E × 75
|n2 + n − 91| (allow the number 1 to be prime) [−24, 23] 48 24 (including the number 1) 45E 4X7 = 17 × 31
|2n2 − 147| (allow the number 1 to be prime) [−23, 23] 47 24 (including the number 1) 88E 961 = 312
|2n2 − 131| [−23, 23] 47 24 8X5 977 = 17 × 61
|4n2 − 16E| [−18, 18] 35 19 965 X81 = 25 × 45
|4n2 − 11E| [−14, 14] 29 15 5E5 6X5 = 1E × 37
|2n2 + 2n − 95| (allow the number 1 to be prime) [−1X, 19] 38 1X (including the number 1) 577 62E = 25 × 27
|2n2 + 2n − 6E| (allow the number 1 to be prime) [−15, 14] 2X 15 (including the number 1) 325 381 = 1E2
|4n2 + 4n − 291| [−23, 22] 46 23 148E 162E = 31 × 5E
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