Le Monde puzzle [#1063]

lemondapariA simple (summertime?!) arithmetic Le Monde mathematical puzzle

  1. A “powerful integer” is such that all its prime divisors are at least with multiplicity 2. Are there two powerful integers in a row, i.e. such that both n and n+1 are powerful?
  2.  Are there odd integers n such that n² – 1 is a powerful integer ?

The first question can be solved by brute force.  Here is a R code that leads to the solution:

isperfz <- function(n){ 
  divz=primeFactors(n) 
  facz=unique(divz) 
  ordz=rep(0,length(facz)) 
  for (i in 1:length(facz)) 
    ordz[i]=sum(divz==facz[i]) 
  return(min(ordz)>1)}

lesperf=NULL
for (t in 4:1e5)
if (isperfz(t)) lesperf=c(lesperf,t)
twinz=lesperf[diff(lesperf)==1]

with solutions 8, 288, 675, 9800, 12167.

The second puzzle means rerunning the code only on integers n²-1…

[1] 8
[1] 288
[1] 675
[1] 9800
[1] 235224
[1] 332928
[1] 1825200
[1] 11309768

except that I cannot exceed n²=10⁸. (The Le Monde puzzles will now stop for a month, just like about everything in France!, and then a new challenge will take place. Stay tuned.)

6 Responses to “Le Monde puzzle [#1063]”

  1. In the solution of the second part, 675 should not be in the list as it is 26^2-1 and 26 is an even number.

  2. […] article was first published on R – Xi’an’s Og, and kindly contributed to […]

  3. FOULLEY Jean-Louis Says:

    Problem probably inspired by the Math exam this year for the Bachelor degree Section S , Math Speciality , Exercise No 4

  4. Seems the first question would be answered by the second one? As n^2 is always powerful, then n^2 - 1 and n^2 are pair integers that satisfy the first question.

    And we can show there are infinitely many: for any n^2 - 1 which is a powerful integer, (2n^2 - 1)^2 - 1 = 4n^2(n^2 - 1) will also be a powerful integer.

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