I'm reading "Friendly Introduction to Number Theory". Now I'm working on Primitive Pythagorean Triples Exercises 2.3 (a) on P19.
2.3. For each of the following questions, begin by compiling some data; next examine the data and formulate a conjecture; and finally try to prove that your conjecture is correct. (But don't worry if you can’t solve every part of this problem; some parts are quite difficult.)
(a) Which odd numbers $a$ can appear in a primitive Pythagorean triple ($a, b, c)$?
https://www.math.brown.edu/~jhs/frintch1ch6.pdf
(1) $a^2 + b^2 = c^2$ with $a$ odd, $b$ even, $a$, $b$, $c$ having no common factors
(2) $a^2 = c^2 - b^2 = (c-b)(c+b)$
(3) $c + b = s^2$ and $c - b = t^2$
(4) $c = \frac{(s^2 + t^2)}{2}$ and $b = \frac{(s^2 - t^2)}{2}$
(5) $a = \sqrt{(c-b)(c+b)} = st$
(6) $a = st$, $b = \frac{(s^2 - t^2)}{2}$, $c = \frac{(s^2 + t^2)}{2}$
I compiled some data and examining it but I can't find the pattern. Can you see any patterns? I need a hint.
https://github.com/y-zono/friendly-introduction-number-theory/blob/master/02/2-3/main.go
{a b c s t}
--------------
{3 4 5 3 1}
{5 12 13 5 1}
{7 24 25 7 1}
{9 40 41 9 1}
{11 60 61 11 1}
{13 84 85 13 1}
{15 8 17 5 3}
{15 112 113 15 1}
{17 144 145 17 1}
{19 180 181 19 1}
{21 20 29 7 3}
{33 56 65 11 3}
{35 12 37 7 5}
{39 80 89 13 3}
{45 28 53 9 5}
{51 140 149 17 3}
{55 48 73 11 5}
{57 176 185 19 3}
{63 16 65 9 7}
{65 72 97 13 5}
{77 36 85 11 7}
{85 132 157 17 5}
{91 60 109 13 7}
{95 168 193 19 5}
{99 20 101 11 9}
a: 3 5 7 9 11 13 15 15 17 19 21 33 35 39 45 51 55 57 63 65 77 85 91 95 99
odd: 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99
Update 1
According davidlowryduda's advice, I compiled some more data. Then I found 23 was appeared.
// max of s = 20
a: 3 5 7 9 11 13 15 15 17 19 21 33 35 39 45 51 55 57 63 65 77 85 91 95 99
odd: 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99
// max of s = 30
a: 3 5 7 9 11 13 15 15 17 19 21 21 23 25 27 29 33 35 39 45 51 55 57 63 65 69 75 77 85 87 91 95 99
odd: 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99
And now 31 is not appeared but I can assume that the number exist when I compile some data more.
$s=31$ and $t=1$ or $s=1$ and $t=31$
$s=31$, $t=1$ then $(s^2−t^2)/2=(961−1)/2=480$ and $(s^2+t^2)/2=(961+1)/2=481$
$31^2+480^2=481^2$
So it turns out that 31 appears.
// max of s = 40
3 5 7 9 11 13 15 15 17 19 21 21 23 25 27 29 31 33 33 35 35 37 39 39 45 51 55 57 63 65 69 75 77 85 87 91 93 95 99
3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99
I confirmed that 31 appeared when I showed the data more. And the 41 is not showed this time..
So I can assume that all odds numbers appear and I think I can find something the pattern.