Euler,s 6n+1 Theorem
المؤلف:
Sloane, N. J. A
المصدر:
Sequences A002476/M4344, A092572, A092573, A092574, and A092575 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
10-10-2020
997
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."
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