Consecutive Primes of form 4n+1

Theorem

The sequence of $16$ consecutive prime numbers beginning from $207 \, 622 \, 273$ are all of the form $4 n + 1$.

Proof

\(\text {(1)}: \quad\)

\(\ds 207 \, 622 \, 273\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 568 + 1\)

and is the $11 \, 477 \, 482$nd prime

\(\text {(2)}: \quad\)

\(\ds 207 \, 622 \, 297\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 574 + 1\)

and is the $11 \, 477 \, 483$rd prime

\(\text {(3)}: \quad\)

\(\ds 207 \, 622 \, 301\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 575 + 1\)

and is the $11 \, 477 \, 484$th prime

\(\text {(4)}: \quad\)

\(\ds 207 \, 622 \, 313\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 578 + 1\)

and is the $11 \, 477 \, 485$th prime

\(\text {(5)}: \quad\)

\(\ds 207 \, 622 \, 321\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 580 + 1\)

and is the $11 \, 477 \, 486$th prime

\(\text {(6)}: \quad\)

\(\ds 207 \, 622 \, 381\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 595 + 1\)

and is the $11 \, 477 \, 487$th prime

\(\text {(7)}: \quad\)

\(\ds 207 \, 622 \, 409\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 602 + 1\)

and is the $11 \, 477 \, 488$th prime

\(\text {(8)}: \quad\)

\(\ds 207 \, 622 \, 417\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 604 + 1\)

and is the $11 \, 477 \, 489$th prime

\(\text {(9)}: \quad\)

\(\ds 207 \, 622 \, 421\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 605 + 1\)

and is the $11 \, 477 \, 490$th prime

\(\text {(10)}: \quad\)

\(\ds 207 \, 622 \, 489\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 622 + 1\)

and is the $11 \, 477 \, 491$st prime

\(\text {(11)}: \quad\)

\(\ds 207 \, 622 \, 501\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 625 + 1\)

and is the $11 \, 477 \, 492$nd prime

\(\text {(12)}: \quad\)

\(\ds 207 \, 622 \, 517\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 629 + 1\)

and is the $11 \, 477 \, 493$rd prime

\(\text {(13)}: \quad\)

\(\ds 207 \, 622 \, 537\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 634 + 1\)

and is the $11 \, 477 \, 494$th prime

\(\text {(14)}: \quad\)

\(\ds 207 \, 622 \, 549\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 637 + 1\)

and is the $11 \, 477 \, 495$th prime

\(\text {(15)}: \quad\)

\(\ds 207 \, 622 \, 553\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 638 + 1\)

and is the $11 \, 477 \, 496$th prime

\(\text {(16)}: \quad\)

\(\ds 207 \, 622 \, 561\)

\(=\)

\(\ds 4 \times 51 \, 905 \, 640 + 1\)

and is the $11 \, 477 \, 497$th prime

Note that the $11 \, 477 \, 481$st prime:

$207 \, 622 \, 271 = 4 \times 51 \, 905 \, 568 - 1$

and the $11 \, 477 \, 498$th prime:

$207 \, 622 \, 567 = 4 \times 51 \, 905 \, 642 - 1$

and so are not of the form $4 n + 1$.

$\blacksquare$

Historical Note

This result is reported by David Wells in his Curious and Interesting Numbers, 2nd ed. of $1997$ as the work of Stephane Vandemergel, but details are lacking.

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $207,622,273$