Euler's Lucky numbers are positive integersnsuch thatm2 −m+nis a prime number form= 0, …,n− 1.

How do we prove that if n is lucky, then n is prime?

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- July 6th 2010, 04:32 PMnoviceEuler's Lucky number
Euler's Lucky numbers are positive integers

*n*such that*m*2 −*m*+*n*is a prime number for*m*= 0, …,*n*− 1.

How do we prove that if n is lucky, then n is prime? - July 6th 2010, 04:59 PMmelese
Try proving this by assuming to the contrary that is composite, or 1.

- July 6th 2010, 05:20 PMAlso sprach Zarathustra
Interesting post!

In my(poor) opinion to prove this we can't do what mr. melese offered...

Here is a link I think it be useful! Conjecture 17. The Ludovicus conjecture about the Euler trinomials - July 6th 2010, 05:20 PMchiph588@
- July 6th 2010, 05:23 PMAlso sprach Zarathustra
- July 6th 2010, 05:24 PMchiph588@
- July 6th 2010, 05:32 PMmelese
- July 6th 2010, 06:07 PMchiph588@
- July 6th 2010, 06:38 PMmelese
But by definition 1 isn't a composite number (nor a prime), so 1 has a unique place.

- July 6th 2010, 06:40 PMchiph588@
- July 6th 2010, 06:51 PMnovice
I found in a different book that .

It said the first 6 lucky numbers are 2, 3, 5, 11, 17 and 41 .

It should make sense to have m=0 since , which is prime. - July 6th 2010, 07:02 PMmelese
- July 6th 2010, 07:03 PMchiph588@
- July 6th 2010, 07:14 PMnovice
I spent half a day trying it many different ways. I attempted to prove this by contradiction at the beginning, but to prove being a composite is rather difficult because there isn't a mathematical expression for prime numbers, so I decided to prove it by contrapositive, but my main difficulty was here:

I let where and and .

Since , I let .

I ended up with , which led me to nowhere, and I decided to post the question.

I found your technique quite interesting. I see why you chose . It makes sense since . That's quite creative. - July 8th 2010, 03:27 PMelim
To show lucky => prime, simply let m = 0

clearly prime numbers are not necessarily lucky (n=7)

Are there infinite many lucky number?