Let the sequence of primes, with $1$ adjoined, be denoted by

Background

All theorems, corollaries, and definitions listed in the book's order:

Theorems and Corollaries in Elementary Number Theory (Ch 1 - 3)

All theorems and corollaries mentioned in David M. Burton’s Elementary Number Theory are listed by following the book’s order. (7th Edition) (Currently Ch 1 - 3)

RanblogRan

I will only use theorems or facts that are proved before this question. So you will not see that I quote theorems or facts from the later chapters.

Question

Let the sequence of primes, with $1$ adjoined, be denoted by $p_{0} = 1, p_{1} = 2, p_{2} = 3, p_{3} = 5, ....$ For each $n \geq 1$, it is known that there exists a suitable choice of coefficients $\epsilon_{k} = \pm 1$ such that

$$ \begin{equation} \begin{split} p_{2n} = p_{2n - 1} + \sum_{k = 0}^{2n - 2} \epsilon_{k}p_{k} \qquad p_{2n + 1} = 2p_{2n} + \sum_{k = 0}^{2n - 1} \epsilon_{k}p_{k} \end{split} \nonumber \end{equation} $$

To illustrate:

$$ 13 = 1 + 2 - 3 - 5 + 7 + 11 $$

and

$$ 17 = 1 + 2 - 3 - 5 + 7 - 11 + 2 \cdot 13 $$

Determine similar representations for the primes $23, 29, 31,$ and $37$.

Solution

(I wrote a program for this question, so I list all possibilities here.)

For $23$:

$p_{9}: 23 = 2 \cdot 19 + 1 + 2 + 3 + 5 - 7 + 11 - 13 - 17$
$p_{9}: 23 = 2 \cdot 19 + 1 - 2 + 3 + 5 - 7 - 11 + 13 - 17$
$p_{9}: 23 = 2 \cdot 19 + 1 - 2 + 3 - 5 + 7 + 11 - 13 - 17$
$p_{9}: 23 = 2 \cdot 19 - 1 + 2 + 3 - 5 - 7 - 11 - 13 + 17$
$p_{9}: 23 = 2 \cdot 19 - 1 + 2 - 3 - 5 + 7 - 11 + 13 - 17$
$p_{9}: 23 = 2 \cdot 19 - 1 - 2 - 3 + 5 - 7 - 11 - 13 + 17$