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Term

Balanced Prime

Primes exactly midway between their prime neighbors (5, 53, 157, 173, 211, 257, 263, 373, …).

434 numbers tagged.

A balanced prime sits exactly at the average of its two prime neighbors: the gap below equals the gap above. The prime 53 is preceded by 47 and followed by 59 — both gaps are 6, so 53 is balanced.

Balanced primes are a window into the fine structure of prime gaps: most primes lean closer to one neighbor than the other, so an exact balance is uncommon. It is conjectured (but open) that infinitely many balanced primes exist. Generalizations look at primes balanced over longer windows of neighbors.

5 53 157 173 211 257 263 373 563 593 607 653 733 947 977 1,103 1,123 1,187 1,223 1,367 1,511 1,747 1,753 1,907 2,287 2,417 2,677 2,903 2,963 3,307 3,313 3,637 3,733 4,013 4,409 4,457 4,597 4,657 4,691 4,993 5,107 5,113 5,303 5,387 5,393 5,563 5,807 6,073 6,263 6,317 6,323 6,367 6,373 6,863 6,977 7,523 7,583 7,823 7,841 8,117 8,713 8,747 9,397 9,473 9,871 10,253 10,607 10,657 10,853 11,299 11,411 11,497 11,731 11,807 11,903 11,933 12,497 12,547 12,583 12,647 12,653 12,841 12,973 13,043 13,177 13,457 13,463 14,543 14,747 14,753 15,161 15,193 15,313 15,467 15,767 15,797 15,803 15,907 15,913 16,097 16,223 16,427 16,487 16,567 16,619 16,787 16,937 16,987 17,047 17,327 17,431 17,477 17,483 17,851 18,217 18,223 18,341 18,433 18,731 19,463 19,477 19,483 19,577 19,603 19,739 19,949 20,107 20,123 20,161 20,201 20,347 20,521 20,731 21,163 21,661 21,911 21,997 22,051 22,073 22,259 22,447 23,327 23,333 23,767 23,801 23,893 24,077 24,097 24,413 24,677 25,391 25,463 25,621 25,673 26,041 26,177 26,183 26,393 26,693 26,723 26,987 27,067 27,773 28,069 28,597 28,927 29,173 29,269 29,333 29,599 29,873 30,059 30,097 30,103 30,313 30,449 30,637 30,643 30,977 31,051 31,327 32,083 32,479 32,497 32,993 33,113 33,857 34,171 34,267 34,543 34,631 34,673 34,877 35,129 35,317 35,753 35,851 35,911 36,067 36,229