Euler's 6n+1 Theorem

Euler's  theorem states that every prime of the form , (i.e., 7, 13, 19, 31, 37, 43, 61, 67, ..., which are also the primes of the form ; OEIS A002476) can be written in the form  with  and  positive integers.

The first few positive integers that can be represented in this form (with ) are 4, 7, 12, 13, 16, 19, ... (OEIS A092572), summarized in the following table together with their representations.

4	(1, 1)
7	(2, 1)
12	(3, 1)
13	(1, 2)
16	(2, 2)
19	(4, 1)
21	(3, 2)
28	(1, 3), (4, 2), (5, 1)
31	(2, 3)

Restricting solutions such that  (i.e.,  and  are relatively prime), the numbers that can be represented as  are 4, 7, 12, 13, 19, 21, 28, 31, 37, 39, 43, ... (OEIS A092574), as summarized in the following table.

	with
4	(1, 1)
7	(2, 1)
12	(3, 1)
13	(1, 2)
19	(4, 1)
21	(3, 2)
28	(1, 3), (5, 1)
31	(2, 3)
37	(5, 2)

REFERENCES:

Sloane, N. J. A. Sequences A002476/M4344, A092572, A092573, A092574, and A092575 in "The On-Line Encyclopedia of Integer Sequences."