**HP Prime, TI-84 Plus CE, Sharp EL-5500 III: Pythagorean Triples**

**Criteria**

The program
PYTHA calculates a Pythagorean triple. A
Pythagorean triple is a set of three positive integers A, B, and C that
represent the lengths of a right triangle, with C being the hypotenuse. Hence, A^2 + B^2 = C^2.

Pythagorean
triples can be generated with three arbitrary positive integers M, N, and K
with the following criteria:

1. M > N

2. M and N are coprime. That is, gcd(M, N) = 1 (gcd, greatest common denominator)

A, B, and C are
generated by:

A = K * (M^2 –
N^2)

B = K * (2 * M
* N)

C = K * (M^2 +
N^2)

**Verification**

We can verify
that the above formulas for A, B, and C work by showing A^2 + B^2 = C^2.

A^2 + B^2

= (K * (M^2 –
N^2))^2 + (K * 2 * M * N)^2

= (K * M^2 – K *
N^2)^2 + 4 * K^2 * M^2 * N^2

= K^2 * M^4 – 2
* K^2 * M^2 * N^2 + K^2 * N^2 + 4 * K^2 * M^2 * N^2

= K^2 * M^4 + 2
* K^2 * M^2 * N^2 + K^2 * N^2

= (K * M^2 + K
* N^2)^2

= K^2 * (M^2 +
N^2)^2

= C^2

**Programs**

**HP Prime Program PYTHA**

EXPORT PYTHA(M,N,K)

BEGIN

// 2017-02-08 EWS

// Pythagorean Triangle

LOCAL A,B,C;

// checks (not for minimum)

M:=IP(M); N:=IP(N); K:=IP(K);

IF M≤0 OR N≤0 OR

gcd(M,N)≠1 OR M≤N THEN

RETURN "INVALID"; KILL;

END;

// calculations

A:=K*(M^2-N^2);

B:=K*(2*M*N);

C:=K*(M^2+N^2);

RETURN {A,B,C};

END;

**TI-84 Plus Program PYTHA**

"EWS
2017-02-08"

Prompt M,N,K

iPart(M)→M

iPart(N)→N

iPart(K)→K

If M≤0 or N≤0 or K≤0
or M≤N or gcd(M,N)≠1

Then

Disp
"INVALID"

Stop

End

K*(M²-N²)→A

K*(2*M*N)→B

K*(M²+N²)→C

Pause {A,B,C}

**Sharp EL-5500 III Pythagorean Triple**

(RUN 450
(line numbers are arbitrary))

450 PAUSE
“Pythagorean Triple”

455 INPUT “M:”;
M, “N:”; N, “K:”; K

460 IF M<=0 OR
N<=0 OR K<=0 OR M<=N THEN 480

462 A = K*(M^2 -
N^2)

464 B = K*2*M*N

466 C = K*(M^2 +
N^2)

468 IF A^2 + B^2
<> C^2 THEN 480

470 PRINT A;
“^2+”; B; “^2=”; C; “^2”

472 END

480 PRINT
“INVALID”

482 END

Examples:

M: 2, N: 1, K:
1. Result: A: 3, B: 4, C: 5

M: 7, N: 2, K:
1. Result: A: 45, B: 28, C: 53

M: 5, N: 3, K:
2. Result: A: 32, B: 60, C: 68

Source:

“Pythagorean
Triple” Wikipedia. Last Modified February 7, 2017.

Accessed
February 7, 2017

Have a great
day, love you all for the support,

Eddie

This blog is
property of Edward Shore, 2017