Highest perfect power of a product of integers less than $x$
Élie Goudout
math.NT 11A05, 11N56




Highest perfect power of a
 product of integers less than x


Élie Goudout



For x≥ 3, we define w(x) as the highest integer w for which there exist integers m, y≥ 1 and 1≤ n_1<…<n_m≤ x such that n_1⋯ n_m=y^w. We show that

    w(x)=xexp(-(√(2)+o(1))√(log xloglog x)).



Keywords: Perfect powers; friable numbers.

MSC: 11A05, 11N56


In <cit.>, Skałba defines, for x≥ 3, w(x) as the highest integer w for which there exist integers m, y≥ 1 and 1≤ n_1<…<n_m≤ x such that n_1⋯ n_m=y^w, and shows

    xexp(-(√(2)+o(1))L(x))≤ w(x)≤ xexp(-(1/√(2)+o(1))L(x)),

where L(x):=√(log xloglog x), as x→+∞. In this paper, we improve the upper bound, making the result optimal up to the exp(o(L(x))).


When x→ +∞,

    w(x)=xexp(-(√(2)+o(1))L(x)).




For 3≤ z≤ x, let Ψ(x,z):=#{n≤ x :   P^+(n)≤ z} be the number of z-friable integers less than x. From <cit.>, we have the classical estimate

    Ψ(x,z)≪ xu^-u+√(x).(u=log x/log z)


From now on, let x≥ 3, m and n_1,…,n_m≤ x denote integers such that n_1⋯ n_m is of the form y^w(x). Let q:=P^+(y^w(x)) denote the highest prime factor of y^w(x) and v_q(y^w(x)) denote its q-valuation. Since the number of q-friable integers n≤ x with q| n is less than Ψ(x/q,q), we have

    w(x)≤ v_q(y^w(w))≤Ψ(x/q,q)log x/log q.

The estimate (<ref>) yields that the maximum of the right member is attained for 

    q=exp((1/√(2)+o(1))L(x)).

In this case, we get the desired upper bound, which, together with Skałba's lower bound, gives the result.

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Institut de Mathématiques de Jussieu-PRG, Université Paris Diderot,
Sorbonne Paris Cité, 75013 Paris, France


E-mail : <eliegoudout@hotmail.com>

