# Four squares theorem etc

We have studied Lagrange's four-square theorem and is denoted by g (2) = 4. i.e., any number can be expressible in sum of squares of four positive integers. Now my question is, here g (2) = 4, where 4 must be reduced to 3 except for numbers of the form $4^n$ (8k + 7). Can you generalize the difficulties, if the number in the form $4^n$ (8k + 7) will trouble to reduce in g (2) = 4 to 3? Also, I am looking the proof or idea to prove g (7) = 143, g (8) = 279, g (9) = 548 and g (10) = 1079.

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Waring's function is not universally known, so you might have included a definition. A starting point for learning more? – Jyrki Lahtonen May 18 '12 at 9:30
@Jyrki Lahtonen! all g(k) can be found from g(k) = $2^k$ + [$(3/2)^k$] -2, which is given in wiki. But, I felt that g(k) can be evaluated for any k, if g(k) = $2^k$ + [$(3/2)^k$] -2 + [$(4/3)^k$]. Can you generalize??? – mr.math May 18 '12 at 9:54
No, I can not! I have never seriously looked at Waring's problem - except in finite fields, where the rules of that game are very different. – Jyrki Lahtonen May 18 '12 at 10:32
I'm sorry, I don't understand "Can you generalize the difficulties?" See also oeis.org/A002804 and the references and links there. – Gerry Myerson May 19 '12 at 9:07