Smallest 22 Primes in Arithmetic Progression

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Theorem

The smallest $22$ primes in arithmetic progression are:

$11 \, 410 \, 337 \, 850 \, 553 + 4 \, 609 \, 098 \, 694 \, 200 n$

for $n = 0, 1, \ldots, 21$.


Proof

\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 0 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 11 \, 410 \, 337 \, 850 \, 553\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 1 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 16 \, 019 \, 436 \, 544 \, 753\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 2 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 20 \, 628 \, 535 \, 238 \, 953\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 3 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 25 \, 237 \, 633 \, 933 \, 153\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 4 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 29 \, 846 \, 732 \, 627 \, 353\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 5 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 34 \, 455 \, 831 \, 321 \, 553\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 6 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 39 \, 064 \, 930 \, 015 \, 753\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 7 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 43 \, 674 \, 028 \, 709 \, 953\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 8 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 48 \, 283 \, 127 \, 404 \, 153\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 9 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 52 \, 892 \, 226 \, 098 \, 353\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 10 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 57 \, 501 \, 324 \, 792 \, 553\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 11 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 62 \, 110 \, 423 \, 486 \, 753\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 12 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 66 \, 719 \, 522 \, 180 \, 953\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 13 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 71 \, 328 \, 620 \, 875 \, 153\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 14 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 75 \, 937 \, 719 \, 569 \, 353\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 15 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 80 \, 546 \, 818 \, 263 \, 553\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 16 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 85 \, 155 \, 916 \, 957 \, 753\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 17 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 89 \, 765 \, 015 \, 651 \, 953\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 18 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 94 \, 374 \, 114 \, 346 \, 153\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 19 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 98 \, 983 \, 213 \, 040 \, 353\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 20 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 103 \, 592 \, 311 \, 734 \, 553\) which is prime
\(\displaystyle 11 \, 410 \, 337 \, 850 \, 553 + 21 \times 4 \, 609 \, 098 \, 694 \, 200\) \(=\) \(\displaystyle 108 \, 201 \, 410 \, 428 \, 753\) which is prime


But note that $11 \, 410 \, 337 \, 850 \, 553 + 22 \times 4 \, 609 \, 098 \, 694 \, 200 = 112 \, 810 \, 509 \, 122 \, 953 = 61 \times 107 \times 1907 \times 9063277$ and so is not prime.



Sources