# table of primes in arithmetic progressions per Dirichlet’s theorem

Dirichlet’s theorem on primes in arithmetic progressions tells us that given the $n$th prime $p_{n}$, there are infinitely many primes of the form $mp_{n}+1$. Obviously, for $p_{1}=2$, the primes of that form are merely the odd primes. For the other primes, $m$ has to be even, but not much else appears obvious.

The leftmost column of this table gives the $n$th prime, the second column from the left gives the smallest prime of the form $mp_{n}+1$, the third column from the left gives the second smallest prime of that form, etc. Apart from the leftmost column, none of the columns contain a sequence in ascending order.

 $p_{n}$ 2 3 5 7 11 13 17 19 23 29 31 3 7 13 19 31 37 43 61 67 73 79 5 11 31 41 61 71 101 131 151 181 191 7 29 43 71 113 127 197 211 239 281 337 11 23 67 89 199 331 353 397 419 463 617 13 53 79 131 157 313 443 521 547 599 677 17 103 137 239 307 409 443 613 647 919 953 19 191 229 419 457 571 647 761 1103 1217 1483 23 47 139 277 461 599 691 829 967 1013 1151 29 59 233 349 523 929 1103 1277 1451 1567 1741 31 311 373 683 1117 1303 1427 1489 1613 1861 2357 37 149 223 593 1259 1481 1777 1999 2221 2591 2887 41 83 739 821 1231 1559 1723 2297 2543 2707 2789 43 173 431 947 1033 1291 1549 1721 1979 2237 2753 47 283 659 941 1129 1223 1693 1787 2069 2351 2539 53 107 743 1061 1697 2333 2969 3181 3499 3923 4241 59 709 827 1063 1181 1889 2243 2833 3187 3541 3659 61 367 733 977 1709 1831 2441 3539 4027 4271 4637 67 269 1609 1877 2011 3083 3217 4021 4289 4423 4691 71 569 853 1279 1847 2131 2273 2557 2699 4261 5113 73 293 439 877 1607 1753 3067 3359 3797 3943 4673 79 317 1423 2213 2371 2687 3319 3793 4583 5531 5689 83 167 499 997 1163 1993 2657 4483 4649 5147 5479 89 179 1069 2137 2671 3739 3917 4273 4451 5519 6053 97 389 971 1553 1747 3299 3881 4463 4657 5821 6791
Title table of primes in arithmetic progressions per Dirichlet’s theorem TableOfPrimesInArithmeticProgressionsPerDirichletsTheorem 2013-03-22 18:13:15 2013-03-22 18:13:15 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Data Structure msc 11N13