Prime to Prime Sequel

Question

This question is inspired by the Prime to Prime puzzle.

The first 24 Prime Numbers are

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89

Using up to 4 prime numbers and the following mathematical operations, get all 24 primes.

+ - × /

No other operators are allowed (note the added restrictions from the previous question).

Other rules

• You cannot use same prime number more than once in any equation.
• You can use only prime numbers.
• You do not have to use all the 4 primes in every equation.
• You must use the same set of 4 primes in every equation. If you select say 2, 13, 17, 23 then they are the only primes that to appear in every equation to get the 24 primes.
• Concatenation is forbidden.
• Parentheses are permitted.

Please refrain from posting partial solutions as there are many sets of primes which will not work.

Answer 1

Here is the answer:

$2=2$

$3=3$

$5=2+3$

$7=\frac{43-3}{2}-13$

$11=43-2\cdot(13+3)$

$13=13$

$17=43-13\cdot2$

$19=13+3\cdot2$

$23=43-2\cdot(13-3)$

$29=43-13-3+2$

$31=43-13+3-2$

$37=43-3\cdot2$

$41=\frac{43+3\cdot13}{2}$

$43=43$

$47=2\cdot43-3\cdot13$

$53=43+13-3$

$59=43+13+3$

$61=43+13+3+2$

$67=2\cdot(43-3)-13$

$71=\frac{3\cdot43+13}{2}$

$73=2\cdot43-13$

$79=2\cdot(43+3)-13$

$83=2\cdot43-3$

$89=2\cdot43+3$

Answer 2

Computer confirms that the only other solution is:

$$\begin{align*}2 &= 2 \\3 &= 3 \\5 &= (2 + 3) \\7 &= - ((3 - 17) / 2) \\11 &= (17 - (2 \cdot 3)) \\13 &= ((17 + 61) / (2 \cdot 3)) \\17 &= 17 \\19 &= (2 + 17) \\23 &= (17 + (2 \cdot 3)) \\29 &= - ((3 - 61) / 2) \\31 &= - (3 - (2 \cdot 17)) \\37 &= (3 + (2 \cdot 17)) \\41 &= (61 - (3 + 17)) \\43 &= ((2 - 3) - (17 - 61)) \\47 &= (61 + (3- 17)) \\53 &= (2 + (3 \cdot 17)) \\59 &= - (2 - 61) \\61 &= 61 \\67 &= (61 + (2 \cdot 3)) \\71 &= ((2 \cdot 61) - (3 \cdot 17)) \\73 &= - ((2 + 3) - (17 + 61)) \\79 &= - ((2 - 3) - (17 + 61)) \\83 &= ((2 + 3) + (17 + 61)) \\89 &= (61 - (2 \cdot (3 - 17)))\end{align*}$$