**A** simple (summertime?!) arithmetic Le Monde mathematical puzzle

*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?*

*Are there odd integers n such that n² – 1 is a powerful integer ?*

**T**he 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.)