<s>[INST] Is there a way to solve for pythagorean triples with one side length equalling 1? When you have one side of a right triangle fixed, is there a trend in pythagorean triples? For instance if I fix one side of a triangle at 1 unit what will the other side have to equal to get pythagorean triplets for the first 100 values? Is there a pattern to this? 
 [/INST] You cannot do it with Pythagorean triples but you can do it with the sum of three cubes.
$$1^3+6^3+8^3=9^3=729$$
$$1^3+71^3+138=144^3=2985984$$
$$1^3+135^3+138^3=172^3=5088448$$
$$1^3+242^3+720^3=729^3=387420489$$
$$1^3+372^3+426^3=505^3=128787625$$
$$1^3+426^3+486^3=577^3=192100033$$
$$1^3+566^3+823^3=904^3=738763264$$
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