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Determine a number that can be expressed as the sum of two positive squares having at least two equivalent forms and also that number should be the lowest. And no repeating positive squares.
2 Answers
- Apratim RLv 61 decade agoFavorite Answer
The smallest such number appears to be 65.
65 = 8² + 1² = 7² + 4².
The indication, not proof, of minimality is explained below.
Suppose a² + b² = c² + d². Then a² - c² = d² - b², i.e.
(a+c)(a-c) = (d+b)(d-b). The "smallest" 4-sequence of pairwise-distinct positive integers (p, q, r, s) such that
pq = rs and p ≡₂ q, r ≡₂ s
is (15, 1, 5, 3), which gives us (a,b,c,d) = (8,1,7,4).
