#math trivia #53 solution

Ten immediate neighbors of 53 aren’t prime:

48 49 50 51 52 … 54 55 56 57 58

The gaps between 47 and 53 and between 53 and 59 tie the “record” for the longest gap between successive primes so far, also achieved by (23,29) and (31,37).  But this is the first time two gaps this long have been adjacent.

According to the Prime Number Theorem, the average gap between primes near a given number n is about ln n where ln n is the natural logarithm of n.  (The Prime Number Theorem states that the density of primes near n is about 1/ln n; the average gap is the inverse of the density.)  Near 53, the average gap according to this formula would be just under 4, so a gap of 6 wouldn’t be unusual.

Although 53 has the most composite neighbors of any prime so far, it doesn’t quite meet the definition of a “lonely prime” given by The On-Line Encyclopedia of Integer Sequences.  In that definition, a prime sets a new record for “loneliness” if and only if the gaps on both sides exceed the corresponding gaps for the previous record-holder.  The sequence of primes with this property starts with 2, 3 (gaps of 1 and 2), 7 (gaps of 2 and 4), 23 (gaps of 4 and 6), and 89 (gaps of 6 and 8).  As noted in the reference, with gaps of 6 and 6, 53 exceeds 23’s gaps on one side but not the other.  But it still seems pretty lonely to me.

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