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
