Pythagorean triple Python

About Pythagorean triple

The Pythagorean theorem, which is the basis of the famous three-square theorem, is in a right triangle. a^2 + b^2 = c^2 Will be established.

The number of combinations in which this (a.b, c) is a natural number is called the Pythagorean triple. The genius Pythagoras, who was active in ancient BC, described this combination as the following formula:

(a,b,c) = \frac{n^2 - 1}{2}, n, \frac{n^2 + 1}{2}

I realized that I could express it as.

However, while this formula can show that there are an infinite number of Pythagorean triples, it is not exhaustive.

So is there a comprehensive formula?

The first answer is said to remain in the 7th century Indian literature.

As m> n a = k(m^2 - n^2), b = 2kmn, c = k(m^2 + n~2)

After all, Indian mathematics is advancing! !!

Output Pythagorean triple in Python

def Pythagoras(n :int) -> list:
    answer = []
    answer.append((n**2-1)/2)
    answer.append(n)
    answer.append((n**2+1)/2)
    return answer
    
for i in range(1,100,2):
    answer = Pythagoras(i)
    for i in range(0,3):
        answer[i] = int(answer[i])
    print(answer)

a	b	c
4	3	5
12	5	13
24	7	25
40	9	41
60	11	61
84	13	85
112	15	113
144	17	145
180	19	181
220	21	221
264	23	265
312	25	313

If it is a small combination, you can see that (6, 8, 10) has been skipped.


# m > n
def Pythagoras_2(k :int, m :int, n: int) -> list:
    answer = []
    answer.append(k*(m**2 - n**2))
    answer.append(2*k*m*n)
    answer.append(k*(m**2 + n**2))
    return answer

for k in range(1,5):
    for m in range(1,5):
        for n in range(1,10):
            if m <= n :
                break
            print(Pythagoras_2(k,m,n))

a	b	c
3	4	5
8	6	10
5	12	13
15	8	17
12	16	20
7	24	25
6	8	10
16	12	20
10	24	26
30	16	34
24	32	40
14	48	50
9	12	15
24	18	30
15	36	39
45	24	51
36	48	60
21	72	75
12	16	20
32	24	40
20	48	52
60	32	68
48	64	80
28	96	100

The combination of Pythagorean triples can be comprehensively expressed.

$ A ^ 2 + b ^ 2 = c ^ 2 $ could not be matched with (a, b, c) in the table output by python. I would appreciate it if you could tell me who can think of good loop processing.

a	b	c
4	3	5
12	5	13
24	7	25
40	9	41
60	11	61
84	13	85
112	15	113
144	17	145
180	19	181
220	21	221
264	23	265
312	25	313

a	b	c
3	4	5
8	6	10
5	12	13
15	8	17
12	16	20
7	24	25
6	8	10
16	12	20
10	24	26
30	16	34
24	32	40
14	48	50
9	12	15
24	18	30
15	36	39
45	24	51
36	48	60
21	72	75
12	16	20
32	24	40
20	48	52
60	32	68
48	64	80
28	96	100

a	b	c
4	3	5
12	5	13
24	7	25
40	9	41
60	11	61
84	13	85
112	15	113
144	17	145
180	19	181
220	21	221
264	23	265
312	25	313

a	b	c
3	4	5
8	6	10
5	12	13
15	8	17
12	16	20
7	24	25
6	8	10
16	12	20
10	24	26
30	16	34
24	32	40
14	48	50
9	12	15
24	18	30
15	36	39
45	24	51
36	48	60
21	72	75
12	16	20
32	24	40
20	48	52
60	32	68
48	64	80
28	96	100

a	b	c
4	3	5
12	5	13
24	7	25
40	9	41
60	11	61
84	13	85
112	15	113
144	17	145
180	19	181
220	21	221
264	23	265
312	25	313

a	b	c
3	4	5
8	6	10
5	12	13
15	8	17
12	16	20
7	24	25
6	8	10
16	12	20
10	24	26
30	16	34
24	32	40
14	48	50
9	12	15
24	18	30
15	36	39
45	24	51
36	48	60
21	72	75
12	16	20
32	24	40
20	48	52
60	32	68
48	64	80
28	96	100