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(This article was first published on <strong><a href="https://xianblog.wordpress.com/2018/08/09/le-monde-puzzle-1063/"> R – Xi'an's Og</a></strong>, and kindly contributed to <a rel="nofollow" href="https://www.r-bloggers.com/">R-bloggers)</a> </div>
<p><strong><img data-orig-file="https://xianblog.files.wordpress.com/2015/10/lemondapari.jpeg?w=456" data-orig-size="275,183" data-medium-file="https://xianblog.files.wordpress.com/2015/10/lemondapari.jpeg?w=456?w=275" data-attachment-id="35411" data-image-meta="{" data-image-description="" src="https://xianblog.files.wordpress.com/2015/10/lemondapari.jpeg?w=456" data-permalink="https://xianblog.wordpress.com/2015/10/25/le-monde-puzzle-934/lemondapari/" data-large-file="https://xianblog.files.wordpress.com/2015/10/lemondapari.jpeg?w=456?w=275" alt="lemondapari" data-image-title="lemondapari" sizes="(max-width: 275px) 100vw, 275px" data-comments-opened="1" class="wp-image-35411" data-recalc-dims="1">A</strong> simple (summertime?!) arithmetic <a rel="nofollow" href="https://xianblog.wordpress.com/2011/09/03/le-monde-puzzle-website/" title="Le Monde puzzle [website]" target="_blank">Le Monde mathematical puzzle</a></p>
<ol>
<li><em>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?<br>
</em></li>
<li><em> Are there odd integers n such that n² – 1 is a powerful integer ?</em></li>
</ol>
<p><strong>T</strong>he first question can be solved by brute force.  Here is a R code that leads to the solution:</p>
<pre>isperfz &lt;- 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]
</pre>
<p>with solutions 8, 288, 675, 9800, 12167.</p>
<p>The second puzzle means rerunning the code only on integers n²-1…</p>
<pre>[1] 8
[1] 288
[1] 675
[1] 9800
[1] 235224
[1] 332928
[1] 1825200
[1] 11309768
</pre>
<p>except that I cannot exceed n²=10⁸. (The <a rel="nofollow" href="https://xianblog.wordpress.com/2011/09/03/le-monde-puzzle-website/" title="Le Monde puzzle [website]" target="_blank">Le Monde</a> puzzles will now stop for a month, just like about everything in France!, and then a <a rel="nofollow" href="https://xianblog.wordpress.com/2017/03/28/le-monde-puzzle-1000-1025/" target="_blank">new challenge</a> will take place. Stay tuned.)</p>
	<p>
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