The 4k-1 versus 4k+1 Prime Race

The number associated with my diurnal age today, 26833, figures in the so-called 4k-1 versus 4k+1 prime race. If we keep a tally of the number of \(4k-1\) primes and \(4k+1\) primes, then eventually certain primes are reached where the two tallies are tied. Here are the list of initial primes where this occurs in the range up to four million:

2, 5, 17, 41, 461, 26833, 26849, 26863, 26881, 26893, 26921, 616769, 616793, 616829, 616843, 616871, 617027, 617257, 617363, 617387, 617411, 617447, 617467, 617473, 617509, 617531, 617579, 617681, 617707, 617719, 618437, 618521, 618593, 618637, 622793, 623171, 623303, 623327, 623351, 623383, 623393, 623431, 623839, 623893, 623929, 623947, 623963, 623983, 624007, 624037, 624049, 624089, 624119, 624139, 624163, 624233, 624251, 626921, 626947, 626959, 627041, 627349, 627383, 627433, 627479, 627491, 627541, 627721, 627811, 627859, 627911, 627973, 628423, 628921, 628939, 629339, 629351, 629381, 629401, 629417, 629483, 629509, 629747, 629773, 630713, 630733, 630823, 630841, 632221, 632561, 632609, 632677, 632717, 632777, 632839, 632881, 632911, 633001, 633091, 633151, 633187, 633427, 633467, 633583, 633613, 633629, 633739, 633757, 633781, 633793, 633799, 633877

As can be seen, 26833 marks the beginning of a cluster of primes (26833, 26849, 26863, 26881, 26893, 26921) after which there is a huge gap until the next cluster (that begins with 616769). Figure 1 shows a graph of the sum of two primes where \(4k-1\) primes are given a value of -1 and \(4k+1\) primes are given a value of +1. When there is an equal number of both types of primes, the sum is zero.

Figure 2: permalink

These primes represent OEIS A007351 and following the link will provides links to other sources of information about this topic.