%I #16 Sep 28 2024 16:16:24

%S 243,70225,265879,953125,1015623,1071873,1922373,6436341,6739605,

%T 7263025

%N Values for b in abc-triples with a = 2.

%C The listed 10 b-values are the ones for all (2,b,2+b) triples

%C with b from the range {1, 2, ..., 10^7}. The best quality among these values appears for n=8: (2, 6436341, 6436343), b = 3^10*109, with rad(2*b*(2+b)) = 15042 =2*3*23*109 and q(2,6436341,6436343) = 1.629911684 (maple 10 digits). See Tabl. I of the (not updated) link: The ABC Conjecture Home Page.

%C See A216323 for the list of increasing b values for abc-triples if a=1. There one finds also a reference and a maple program which can be adapted to a=2 instead of a=1.

%C This sequence is infinite because it contains the infinite subsequence b(k) = 243^(84k+1), k >= 0. - _William Hu_, Aug 29 2024

%H The ABC Conjecture Home Page, <a href="https://nitaj.users.lmno.cnrs.fr/abc.html#Ten%20abc">The top ten good abc-examples</a>.

%F (2, b=a(n), 2+a(n)) is an abc-triple (which has quality q > 1) with increasingly ordered b values. See the comment above for abc-triples.

%e n: (a=2, b, c=2+a), rad(a*b*c), q(a*b*c) (maple 10 digits)

%e 1: (2, 243, 245), 210, 1.028828797

%e 2: (2, 70225, 70227), 27030, 1.093563284

%e 3: (2, 255879, 255881), 252642, 1.001024059

%e 4: (2, 953125, 953127), 525210, 1.045245231

%e 5: (2, 1015623, 1015625), 128310, 1.175886268

%e 6: (2, 1071873, 1071875), 926310, 1.010623492

%e 7: (2, 1922373, 1922375), 799890, 1.064510569

%e 8: (2, 6436341, 6436343), 15042, 1.629911684

%e 9: (2, 6739605, 6739607), 3621030, 1.041135746

%e 10: (2, 7263025, 7263027), 94710, 1.378732296

%e ...

%e From _Wolfdieter Lang_, Oct 02 2012: (Start)

%e The prime number decomposition of the ten b-values is

%e 3^5, 5^2*53^2, 3^9*13, 5^6*61, 3^2*7^4*47, 3^5*11*401, 3^8*293, 3^10*109, 3^6*5*43^2, 5^2*7^4*11^2.

%e The ten c = b+2 numbers have the prime number decomposition

%e 5*7^2, 3^5*17^2, 41*79^2, 3^4*7*41^2, 5^7*13, 5^5*7^3, 5^3*7*13^3, 23^5, 7^5*401, 3^11*41. (End)

%p See the program given in A216323, adapted to a=2.

%Y Cf. A216323, A216370.

%K nonn,more

%O 1,1

%A _Wolfdieter Lang_, Sep 28 2012