Additive conditions on sums of squares
Abstract
Functions F: ℕ→ℝ which satisfy the condition
∀x,y,u,v (ax2+cy2 = au2+cv2 ⇒ a·F(x)+c·F(y) = a·F(u)+c·F(v))
are considered. For some small positive integers a, c they are completely characterized. E.g., for a = c = 1 they form a σ-dimensional real vector space with a base consisting of F0(x) = x2 and five periodical functions with periods 1,..., 5. Further, functions F: ℕ→ℝ which satisfy the condition
∀x,y,z (x2 = y2+z2 ⇒ F(x) = F(y)+F(z))
are studied; 17 linearly independent examples of such functions are presented, including periodical ones with the periods 16, 9, 25, 13.
Keywords
square-additive functions
References
2. I. Korec, Definability of arithmetical operations from binary quadratic forms, Preprint 5/1998 of Math. Institute SAV Bratislava, 10.
3. A. Schinzel, On Pythagorean triangles, Ann. Math. Silesianae, 12 (1998), 31.
Mathematical Institute, Slovak Academy of Sciences Slovakia
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