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The largest prime factor of $$X^3+2$$. (English) Zbl 1384.11093
It was shown by the reviewer [Proc. Lond. Math. Soc. (3) 82, No. 3, 554–596 (2001; Zbl 1023.11048)] that there is a positive constant $$\varpi$$ such that one has $P(n^3+2)>n^{1+\varpi}$ for a positive proportion of integers $$n$$ in any sufficiently long range $$(X,2X]$$. Here $$P(m)$$ denotes the largest prime factor of the integer $$m$$. The reviewer’s proof produces $$\varpi=10^{-303}$$, which is disappointingly small.
The present paper shows how to improve one part of the reviewer’s argument, and allows one to take $$\varpi=10^{-52}$$. Unfortunately this is still rather small.
The paper focuses on the quantity $$T(h,\delta;X)$$, which is the number of integers $$n\in(X,2X]$$ for which $$n^3+2$$ has at least $$h$$ prime factors $$p>X^{\delta}$$. Roughly speaking, the paper shows that $$T(h,\delta;X)\leq 2^{-h}X$$ for the range of parameters under consideration, provided that $$h$$ is larger than a certain multiple of $$\log(1/\delta)$$. In effect this provides a bound for the key quantity $\sum_h \min(h,\delta^{-1})T(h,\delta;X)$ which one may think of as being polynomial in $$\delta$$, rather than exponential as in the reviewer’s work. However one eventually works with $$\delta=1/321$$, so that such an asymptotic interpretation needs to be treated with caution.
The paper is necessarily quite technical, but would be a good starting point for anyone hoping to obtain a “realistic” value for $$\varpi$$.
##### MSC:
 11N32 Primes represented by polynomials; other multiplicative structures of polynomial values 11N25 Distribution of integers with specified multiplicative constraints
##### Keywords:
prime factor; polynomial; cubic; largest prime factor; exponent
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##### References:
 [1] [1]P. Billingsley, On the distribution of large prime divisors, Period. Math. Hungar. 2 (1972), 283–289. · Zbl 0242.10033 [2] [2]R. de la Bret‘eche, Plus grand facteur premier de valeurs de polyn\hat{}omes aux entiers, Acta Arith. 169 (2015), 221–250. · Zbl 1379.11083 [3] [3]C. Dartyge, Le probl‘eme de Tch\'{}ebychev pour le douzi‘eme polyn\hat{}ome cyclotomique, Proc. London Math. Soc. (3) 111 (2015), 1–62. 80 · Zbl 1323.11069
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