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I've noticed that for the smaller primes, it is possible to state each element of its reduced residue class as a simple equation in terms of the factors of the primorial.

For example, consider the primorial $5\# = 30$

The reduced residue class is $\{1,7,11,13,17,19,23,29\}$

Each of these elements can be expressed as a simple equation in terms of the factors of the primorial.

  • 1 = $\frac{30}{5} - 5$
  • 7 = $\frac{30}{3} - 3$
  • 11 = $\frac{30}{5} + 5$
  • 13 = $\frac{30}{2} - 2$
  • 17 = $\frac{30}{2} + 2$
  • 19 = $\frac{30}{2} + 2*2$
  • 23 = $\frac{30}{2} + 2*2*2$
  • 29 = $30 - 1$

Is this always the case? Is there a point where this stops being true?

Matthew Conroy
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Larry Freeman
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    This may depend a lot on what you consider to be "a simple equation". – Gerry Myerson Dec 31 '13 at 22:13
  • By "simple", I meant something like this: $\frac{a}{b} + c$ where $a,b,c$ are all expressible solely in terms of factors of $p#$. – Larry Freeman Dec 31 '13 at 22:14
  • Now it depends on what you mean by "expressible solely in terms of factors of $p#$". – Gerry Myerson Dec 31 '13 at 22:20
  • By "expressible solely in terms of factors $p#$", I just meant that each $a,b,c$ is in the form of $f_i^{n_i}f_{i+1}^{n_{i+1}}\cdots$ and each $f_i | p#$. – Larry Freeman Jan 01 '14 at 01:05
  • @LarryFreeman Is there a reason you don't just say $a \pm b$, where $a$ and $b$ only have prime factors dividing $n#$? Or do you want (your) $a/b$ to be square-free? – Matthew Conroy Jan 01 '14 at 01:23
  • Hi Matthew, As you point out, I mean $a \pm b$ where $a$ and $b$ have only prime factors dividing $n#$. I used $\frac{a}{b}$ solely because I wanted to make it as clear as possible that $\frac{30}{2} + 2$ could not be divisible by $2, 3,$ or $5$. – Larry Freeman Jan 01 '14 at 08:33

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