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Date: 2-5-2020
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Date: 8-1-2020
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A Pythagorean quadruple is a set of positive integers , , , and that satisfy
(1) |
For positive even and , there exist such integers and ; for positive odd and , no such integers exist (Oliverio 1996).
Examples of primitive Pythagorean quadruples include , , , , , and .
Oliverio (1996) gives the following generalization of this result. Let , where are integers, and let be the number of odd integers in . Then iff (mod 4), there exist integers and such that
(2) |
A set of Pythagorean quadruples is given by
(3) |
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(4) |
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(5) |
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(6) |
where , , and are integers (Mordell 1969). This does not, however, generate all solutions. For instance, it excludes (36, 8, 3, 37).
REFERENCES:
Carmichael, R. D. Diophantine Analysis. New York: Wiley, 1915.
Dutch, S. "Power Page: Pythagorean Quartets." https://www.uwgb.edu/dutchs/RECMATH/rmpowers.htm#pythquart.
Mordell, L. J. Diophantine Equations. London: Academic Press, 1969.
Oliverio, P. "Self-Generating Pythagorean Quadruples and -tuples." Fib. Quart. 34, 98-101, 1996.
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كيف تساهم الأطعمة فائقة المعالجة في تفاقم مرض يصيب الأمعاء؟
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المجمع العلمي يختتم دورته القرآنية في فن الصوت والنغم بالطريقة المصرية
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