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The largest prime factor of X 3 +2
Author(s) -
HeathBrown D. R.
Publication year - 2001
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/82.3.554
Subject(s) - mathematics , ramanujan's sum , kloosterman sum , conjecture , combinatorics , prime (order theory) , prime factor , constant (computer programming) , order (exchange) , moduli , upper and lower bounds , discrete mathematics , mathematical analysis , physics , finance , quantum mechanics , computer science , economics , programming language
The largest prime factor of X 3 +2 was investigated in 1978 by Hooley, who gave a conditional proo that it is infinitely often at least as large as X 1 +δ, with a certain positive constant δ. It is trivial to obtain such a result with δ=0. One may think of Hooley's result as an approximation to the conjecture that X 3 +2 is infinitely often prime. The condition required by Hooley, his R * conjecture, gives a non‐trivial bound for short Ramanujan–Kloosterman sums. The present paper gives an unconditional proof that the largest prime factor of X 3 +2 is infinitely often at least as large as X 1+δ , though with a much smaller constant than that obtained by Hooley. In order to do this we prove a non‐trivial bound for short Ramanujan–Kloosterman sums with smooth modulus. It is also necessary to modify the Chebychev method, as used by Hooley, so as to ensure that the sums that occur do indeed have a sufficiently smooth modulus. 2000 Mathematics Subject Classification : 11N32.