Goldbach-Zhou conjecture

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The Goldbach-Zhou conjecture is a stricter version of the Goldbach conjecture for a smaller class of numbers.

Conjecture. Any singly even number greater than 2 can be expressed as the sum of two primes, not necessarily distinct, of the form $4k+3$.

Since a singly even number is congruent to 2 modulo 4, two primes congruent to 3 modulo 4 will add up to a singly even number. But how do we prove that among the potential pairs for a singly even number there is always at least one pair of primes?

Lei Zhou has checked the conjecture up to 10⁹. Though slower than in the original conjecture, as $n$ gets larger, the number of potentially suitable representations increases, and it appears unlikely that they would all fail. But of course a preponderance of evidence is no substitute for a rigorous proof.

The following table gives the Goldbach-Zhou representations of singly even numbers up to 98. The representations are sorted in ascending order according to the smaller number of the pair. The smallest numbers in the second column form the sequence A214834.

6	3 + 3
10	3 + 7
14	3 + 11 = 7 + 7
18	7 + 11
22	3 + 19 = 11 + 11
26	3 + 23 = 7 + 19
30	7 + 23 = 11 + 19
34	3 + 31 = 11 + 23
38	7 + 31 = 19 + 19
42	11 + 31 = 19 + 23
46	3 + 43 = 23 + 23
50	3 + 47 = 7 + 43 = 19 + 31
54	7 + 47 = 11 + 43
58	11 + 47
62	3 + 59 = 19 + 43 = 31 + 31
66	7 + 59 = 19 + 47 = 23 + 43
70	3 + 67 = 11 + 59 = 23 + 47
74	3 + 71 = 7 + 67 = 31 + 43
78	7 + 71 = 11 + 67 = 19 + 59 = 31 + 47
82	3 + 79 = 11 + 71 = 23 + 59
86	3 + 83 = 7 + 79 = 19 + 67 = 43 + 43
90	7 + 83 = 11 + 79 = 19 + 71 = 23 + 67 = 31 + 59 = 43 + 47
94	11 + 83 = 23 + 71 = 47 + 47
98	19 + 79 = 31 + 67

Singly even numbers greater than 10 can also be represented as the sum of two primes congruent to 1 modulo 4, and much of what has been said for Goldbach-Zhou representations applies equally well. As for multiples of 4, their Goldbach representations must consist of a prime congruent to 1 modulo 4 and the other congruent to 3.