## Archive for Montpellier

## domaine de l’Hortus [blanc]

Posted in Statistics with tags Chardonnay, Dolines de l'Hortus, French wines, Grande Cuvée, Hortus, Languedoc wines, Montpellier, Petit Manseng, Pic Saint Loup, Saint-Martin-de-Londres, Sauvignon Gris, Val de Montferrand, Viognier on July 20, 2020 by xi'an## Le Monde puzzle [#1083]

Posted in Books, Kids, R, Travel with tags arithmetics, flight, Le Monde, mathematical puzzle, Montpellier, R, Université de Montpellier on February 7, 2019 by xi'an**A **Le Monde mathematical puzzle that seems hard to solve without the backup of a computer (and just simple enough to code on a flight to Montpellier):

Given the number N=2,019, find a decomposition of N as a sum of non-trivial powers of integers such that (a) the number of integers in the sum is maximal or (b) all powers are equal to 4. Is it possible to write N as a sum of two powers?

It is straightforward to identify all possible terms in these sums by listing all powers of integers less than N

pool=(1:trunc(sqrt(2019)))^2 for (pow in 3:11) pool=unique(c(pool,(2:trunc(2019^(1/pow)))^pow))

which leads to 57 distinct powers. Sampling at random from this collection at random produces a sum of 21 perfect powers:

1+4+8+9+16+25+27+32+36+49+64+81+100+121+125+128+144+169+196+243+441

But looking at the 22 smallest numbers in the pool of powers leads to 2019, which is a sure answer. Restricting the terms to powers of 4 leads to the sequence

1⁴+2⁴+3⁴+5⁴+6⁴ = 2019

And starting from the pools of all possible powers in a decomposition of 2019 as the sum of two powers shows this is impossible.