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Yes, this has infinitely many solutions. Let $t$ be any integer, and simply take $a = t+1$, $b = 2^t-1$ and $c = 2t$. Your example is precisely the case $t=4$.

At the risk of being redundant and repeating earlier answers, let me mention that this is explicitly contained in the book "Elementary number theory, group theory and Ramanujan graphs" by Davidoff, Sa …

The ring $\overline{\mathbb{Z}}$ is called the ring of algebraic integers. You can find information about prime ideals, e.g., in https://math.stackexchange.com/questions/156231/non-zero-prime-ideals-i …

This is related to Linnik's theorem: http://en.wikipedia.org/wiki/Linnik%27s_theorem . See in particular the conjecture on this wikipedia page: It is also conjectured that: $p(a,d) < d^2$, where …

There are many possible answers to this question, and it depends a lot on the context, but certainly one of the main reasons is that you cannot distinguish between $1$ and $-1$ modulo $2$, whereas $1 …

The uniqueness of the finite Ree groups of type $^2G_2$ was established by E. Bombieri using extremely tricky number theoretical methods (involving involved elimination methods). As Stephen D. Smith w …

The Algebraic and Geometric Theory of Quadratic Forms by Elman, Karpenko and Merkurjev is a standard recent reference for the theory of quadratic forms, paying special attention to the differences bet …

For any positive integer $n$, we define $$\sigma(n) := \sum_{d \mid n} d,$$ and $$\delta(n) := \frac{\sigma(n)}{n} = \sum_{d \mid n} \frac{1}{d}.$$ Is there an (efficient) way to determine $\delta^{-1 …

Your sequences are simply the coefficients of the polynomials $$(1 + x + x^2 + \dots + x^9)^L .$$ (This is a straightforward application of the theory of generating functions.)