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Sums of powers with prime exponents, summing a square

Source: Problem 4, Brazilian MO, 2012

November 28, 2012

floor functionnumber theory unsolvednumber theory

Problem Statement

There exists some integers

n,a_1,a_2,\ldots,a_{2012}

such that

n^2=\sum_{1 \leq i \leq 2012}{{a_i}^{p_i}}

where

p_i

is the i-th prime (

p_1=2,p_2=3,p_3=5,p_4=7,\ldots

) and

a_i>1

for all

i