Pythagorean Triples

This program finds and displays Pythagorean triples — sets of integers (X, Y, H) satisfying X²+Y²=H² — by iterating X from 1 to 75 and Y from X to 100. The hypotenuse candidate is computed as the square root of X²+Y², with a small epsilon offset (0.00000004) added before the integrality test to compensate for floating-point rounding errors. When a valid triple is found, it is printed and the display scrolls every two results to keep output readable, with a long PAUSE (40000 counts) between findings. The program also includes an inline SAVE command at line 9998 with a filename containing inverse-video characters, and a GOTO loop back to line 10 at line 9999.

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Program Analysis

Program Structure

The program is organized into a simple sequential flow with two nested FOR loops as its core engine:

1. Line 0: Title/author banner in inverse video (display only, never executed as code).
2. Lines 10–90: Initialization — reset counter A, clear screen, print heading.
3. Lines 100–200: Main computation loop — nested FOR X / FOR Y with triple detection and display.
4. Line 9997: STOP to halt after all iterations.
5. Line 9998: SAVE "SIDE%S" — saves the program; only reached manually or in certain flow paths.
6. Line 9999: GOTO 10 — restart loop (unreachable in normal execution after STOP).

Algorithm

The program searches for integer solutions to the Pythagorean equation X²+Y²=H² by brute force. The outer loop runs X from 1 to 75, and the inner loop runs Y from X to 100 (avoiding duplicate pairs where X>Y). For each pair, line 130 computes the candidate hypotenuse:

LET H=(X*X+Y*Y)**.5+.00000004

The fractional part of H is then tested at line 140: if H-INT H > 0.0000001, the triple is rejected. If the condition is false (i.e., H is effectively an integer), the triple is printed.

Floating-Point Epsilon Technique

The small constant 0.00000004 added to the square root in line 130 is a deliberate epsilon offset. Because floating-point square roots of perfect squares may fall just below the true integer value (e.g., 4.9999999 instead of 5.0), the offset nudges the result upward so that INT H correctly rounds to the integer value. The rejection threshold of 0.0000001 in line 140 is chosen to be large enough to pass near-integer results yet small enough to reject genuinely non-integer hypotenuses. This two-constant approach is a well-known idiom for integer detection via floating-point arithmetic on 8-bit systems.

Display and Pacing

Results are printed using the comma separator in line 150 (PRINT X;" ";Y;" ";H,), which advances to the next print zone rather than a new line, allowing compact output. The counter A tracks the number of triples found; line 170 scrolls the display after every second result using the even/odd test IF A/2=INT(A/2). A PAUSE 40000 at line 180 inserts a long delay after each triple is displayed, giving the user time to read before the next result appears.

Key Variables

Variable	Role
A	Count of triples found; used to trigger SCROLL every two triples
X	Shorter leg of the candidate triple (1–75)
Y	Longer or equal leg of the candidate triple (X–100)
H	Candidate hypotenuse (floating-point square root with epsilon)

Notable Techniques and Idioms

• Starting the inner loop at Y=X rather than Y=1 avoids printing both (3,4,5) and (4,3,5) as separate results.
• The even/odd scroll trigger (A/2=INT(A/2)) is a standard modulo-2 idiom on platforms lacking a MOD operator.
• The SAVE command at line 9998 uses an inverse-video filename (SIDE%S), where %S represents an inverse-video “S”. This is a cosmetic choice and does not affect the save operation functionally.
• Line 0 is used as a non-executing banner line. The ZX81/TS1000 will never reach line 0 from a GOTO 10, so it serves purely as onscreen documentation when the program is listed.