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 (Euler
1763; Ribenboim 1979, p. 4).
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) |
