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A216370 Number of ABC triples with quality q > 1 and c < 10^n. 2
1, 6, 31, 120, 418, 1268, 3499, 8987, 22316, 51677, 116978, 252856, 528275, 1075319, 2131671, 4119410, 7801334, 14482059 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The abc conjecture (or Oesterlé-Masser conjecture) is a conjecture in number theory, first proposed by Joseph Oesterlé and David Masser in 1985, stated in terms of three positive integers, a, b and c (whence the name), which have no common factor and satisfy a + b = c. Quantity q = log(c)/log(rad(a*b*c)) where rad(k) = A007947(k), is the product of the distinct prime factors of k. This is the q > 1 column of the wikipedia table taken from RekenMeeMetABC.nl (2011). Goldfeld (1996) described the abc conjecture as "the most important unsolved problem in Diophantine analysis." Jordan Ellenberg at Quomodocumque reports on a potential breakthrough in number theory, a claimed proof of the abc conjecture by Shin Mochizuki.

REFERENCES

Dorian Goldfeld, Beyond the last theorem, Math Horizons, 1996 (September), pp. 26-34.

Richard K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 2004, ISBN 0-387-20860-7.

Carl Pomerance, Computational Number Theory, The Princeton Companion to Mathematics, Princeton University Press, 2008, pp. 361-362.

LINKS

Table of n, a(n) for n=1..18.

Jordan Ellenberg, Mochizuki on ABC

Reken Mee met ABC, Synthese resultaten, (Dutch), 2011

Wikipedia, abc conjecture

EXAMPLE

a(2) = 6 because there are 6 (a,b,c) triples with c < 10^2 and q > 1. Those triples are {1,8,9}, {1,48,49}, {1,63,64}, {1,80,81}, {5,27,32}, and {32,49,81}.

MATHEMATICA

rad[n_] := Times @@ Transpose[FactorInteger[n]][[1]]; Table[t = {}; mx = 10^n; Do[c = a + b; If[c < mx && GCD[a, b] == 1 && Log[c] > Log[rad[a*b*c]], AppendTo[t, {a, b, c}]], {a, mx/2}, {b, a, mx - a}]; Length[t], {n, 3}] (* T. D. Noe, Sep 06 2012 *)

CROSSREFS

Cf. A007947, A120498, A130510, A130511, A130512.

Sequence in context: A143568 A166786 A003728 * A225425 A128568 A079924

Adjacent sequences:  A216367 A216368 A216369 * A216371 A216372 A216373

KEYWORD

nonn

AUTHOR

Jonathan Vos Post, Sep 05 2012

STATUS

approved

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Last modified April 21 11:59 EDT 2014. Contains 240824 sequences.