My question is about the nature of primes as you go to infinity.
I was watching a video about the last digits of primes and Chebeshev's bias and I had a thought about the Goldbach conjecture. If N ends with a 2, then N minus primes ending in 7 will equal a multiple of 5. The same can be done to find multiples of 7 in base 14 (minus multiples of 5 of course) and so forth.
I used this with some code to predict how many prime pairs would add up to N where N has only simple factors of 2's, 3's and primes greater than the square root of N/2 . My laptop could only handle this up to just above 1 billion.
That biggest calculation had a prediction that was about half of the actual result and smaller numbers had a smaller difference. The predictions were always smaller that the actual amount.
I know this can't hold as we go to infinity but why not? All I can think of is that Chebeshev's bias must become much much higher than 3/1000 but that contradicts the video I watched.
Here are the results just for reference.
| N (multiple of) | Prediction(Pre) | Actual | Pre/Actual |
|---|---|---|---|
| 54 (3,2) | 5.2 | 6 | 0.866666667 |
| 92 (Px4) | 3.7 | 4 | 0.925 |
| 128 (2) | 4.1 | 5 | 0.82 |
| 162 (3,2) | 9.4 | 10 | 0.94 |
| 212 (Px4) | 5.6 | 7 | 0.8 |
| 486 (3,2) | 20.1 | 24 | 0.8375 |
| 1,024 (2) | 17.1 | 23 | 0.743478261 |
| 1,458 (3,2) | 44.4 | 48 | 0.925 |
| 4,096 (2) | 47.8 | 53 | 0.901886792 |
| 6,088 (Px8) | 64.1 | 71 | 0.902816901 |
| 39,366 (3,2) | 558.7 | 569 | 0.981898067 |
| 65,536 (2) | 419.6 | 438 | 0.957990868 |
| 112,396 (Px4) | 655.5 | 672 | 0.975446429 |
| 524,288 (2) | 2,335.9 | 2,372 | 0.984780776 |
| 1,062,882 (3,2) | 8,421.2 | 8,607 | 0.97841292 |
| 1,495,636 (Px4) | 5,608.2 | 5,711 | 0.981999649 |
| 4,194,304 (2) | 13,319.9 | 13715 | 0.971192125 |
| 9,565,938 (3,2) | 52,912.6 | 55,737 | 0.9493263 |
| 16,489,952 (Px32) | 41,427.4 | 44,863 | 0.923420102 |
| 33,554,432 (2) | 74,058.4 | 83,480 | 0.887139435 |
| 86,093,442 (3,2) | 313,306.8 | 382,818 | 0.818422331 |
| 155,140,352 (Px256) | 245,376.7 | 322,551 | 0.760737682 |
| 258,280,326 (3,2) | 712,371.8 | 1,015,231 | 0.701684444 |
| 536,870,912 (2) | 585,543.5 | 975,734 | 0.600105664 |
| 1,073,741,824 (2) | 889,644.5 | 1,817,166 | 0.489578002 |