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The definition of ec-numbers is here :

A conjecture about numbers of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$.

I use a slightly different notation :

Suppose $$x=2^n-1$$ and $$y=10^m$$ where $m$ is the number of digits in the decimal expansion of $x$. Then, the "ec-number" is $$2xy+x+y$$ I figured out that $$(x+3y+1)^2-(-x+y+1)^2-8y^2=4(2xy+x+y)$$ holds, hence for a prime factor $q$ of the "ec-number" , we have $$8y^2+(-x+y+1)^2\equiv (x+3y+1)^2\mod q$$

Is this representation of the ec-number useful to rule out specific "ec-numbers" to be prime ? Can we reduce the number of candidates significantly with this expression to accelerate the search for further primes ?

Peter
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  • Maybe is useful in this post add some tags related to (quadratic-forms) if you think that it is important in your approach about your representation or primality issues are related in a direct way to quadratic forms. –  Apr 09 '18 at 09:58

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