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where $X$ is an odd prime, and $a$ is an odd integer.

For example, let $X = 37$, $a = 3$, we get $$\frac{37^3-1}{36} = 3 \times 7 \times 67.$$ When factoring numbers such as this, I've noticed that almost all have at least one prime factor larger than $X$ (e.g. 67 > 37). I would like to know for what values of $X$, $a$ are ALL of the prime factors of $(X^a-1)/(X-1)$ less than $X$. For example, let $X = 79$, $a = 3$, we get $$\frac{79^3-1}{78} = 3 \times 7^2 \times 43$$ and $43 < 79$.

My math education level is first year of high school so a transparent explanation, if possible, would be great. I understand basic congruences.

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    It's worth pointing out that since all the factors of X-1 are less than X (trivially), what you're really asking is when all the prime factors of X^a-1 are less than X. Unfortunately, primes and factoring tend to behave 'without rhyme or reason' when it comes to additivity properties (things like relating the factorizations of X+1 and X), so there's not much reason to believe that there's any structure at all to the answers.2012-05-29
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    $\TeX$ will enlighten your question, I believe.2012-05-29
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    [Cyclotomic polynomials](http://en.wikipedia.org/wiki/Cyclotomic_polynomial) might be relevant.2012-05-29
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    another relevant aspect is, whether the exponent $a$ is prime or composite. If it is prime, then the number of primefactors of the expression is small and thus the factors themselves must be large. If $a$ is composite, then the number of involved primefactors is bigger and thus the involved primefactors are smaller.2012-05-30

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