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Are Euclid numbers squarefree ? An Euclid number is by definition a Primorial number + 1. See http://mathworld.wolfram.com/Primorial.html.

In notation the $n$ th Euclid number is written as $E_n = P_n+1.$

Thus I wonder about $a^2b = E_n$ for positive integer $a,b,n$ and $a>1$.

I was thinking about Korselt's criterion : http://mathworld.wolfram.com/KorseltsCriterion.html and Fermats little.

Im unaware of other techniques for proving the squarefree property apart from more elementary tools as gcd and basic modular aritmetic.

I cannot imagine infinite descent to work here ? Maybe a binomium will help ?

I guess I missed something trivial and have a bad day.

Maybe $a^2b = E_n-2$ is easier ?

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    How far do you have numerical evidence? My quick checks started to slow down already at $197\cdot193\cdot\ldots\cdot 2+1$.2012-10-24
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    Euclid said if you take any finite set of prime numbers and take their product (actually Euclid said their LCM), and add $1$, the prime factors of the resulting number are not in the finite set you started with. If the finite set you started with is $\{3\}$, then the number you get is $4$, and that's not squarefree. If you start with $\{3,5\}$, you get $16$, which is not squarefree. And with $\{5,7\}$ you get $36$. (But it seems to be conventional today to define "Euclid numbers" by taking the initial set of primes to contain all primes less than some number.)2012-10-24
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    I haven't thought about it much, but I'd be surprised if there is an algebraic reason for this conjecture to be true. But a pseudo-statistical argument suggests there is a non-zero probability that it could be "randomly" true.2012-10-24
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    @MichaelHardy : I note that 4 is 2*2 and 2 < 3 , 16 = 4*4 and 4 < 5 and 36 = 6*6 and 6 < 7. So that is far from a counterargument , I dare say the opposite. Also you simply used (x-1)(x+1) = x^2 - 1.2012-10-24
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    @mick : I thought I was pretty clear that what I wrote was not intended as a "counterargument". What relevance you intend the fact that $6<7$, etc., to have, escapes me.2012-10-24
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    @mick: Michael wrote an article on Euclid's proof of the infinitude of primes, which is often misrepresented in the literature. I think he was mostly trying to point out that the name "Euclid numbers" given to these numbers today shouldn't be taken to indicate that they figure in Euclid's proof. Also, one might expect any systematic arithmetic property of the Euclid numbers to be shared by the numbers Michael mentioned, so it wasn't a counterexample, but a reason to doubt that there's a systematic reason for your conjecture, even if it, as Hurkyl said it might, turns out to be "randomly" true.2012-10-24
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    @MichaelHardy : Well $6 < 7$ and $6=3*2$ but $3$ and $2$ are primes smaller than $7$. If you started $2*3*5*7 (+1)$ you would not get 0 mod 6.2012-10-24
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    As for the name ' Euclid number' It was named after Euclid many years after his death , in resemblance(!) of his proof. Rather than being exaclty part of his proof , or being invented by Euclid himself. Ofcourse a proof involving the Euclid numbers is very similar and works very well to show the infinitude of primes. In fact this is more convincing to young students. And imho it has some beauty.2012-10-24
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    @Hurkyl: I posted a heuristic calculation of the "probability" -- is this roughly what you had in mind?2012-10-25
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    @joriki: Yes, that is the approach I had in mind. +1!2012-10-25
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    On page 525 of Graham, Knuth, Patashnik, Concrete Mathematics, it says no square less than $25\times10^{14}$ divides a Euclid number.2012-10-25
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    Page 510 of the downloadable version.2012-10-25
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    Thank you @joriki for pointing out the paper written by Michael Hardy. It was an interesting read. I will have to be careful of how I continue to present Euclid's proof.2015-05-22

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