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Question Number 212584 by MrGaster last updated on 18/Oct/24


Commented by MrGaster last updated on 18/Oct/24
I can't modify and divide this post by three, so I will state the topic in the form of comments: At least how many squares (including 0) sum can represent all natural numbers. If you choose at random, it seems that 4 is the final answer, and most of them are 8n+7. Now please prove that 8n+7 cannot be written as the sum of three squares 2.Excluding these 8n+7, there are still four numbers left, and I find that they are related to 8n+7: they can be written in the form of 4 a * (8n+7). Please prove that all 4 a* (8n+7) cannot be written in the form of the sum of at least three squares.
	Answered by Frix last updated on 18/Oct/24
	$8n+7=u^2 +v^2 +w^2 8n+7 is odd ⇒ { ((u=2j+1∧v=2k∧w=2l)),((u=2j+1∧v=2k+1∧w=2k+1)) :} (1) 8n+7=(2j+1)^2 +(2k)^2 +(2l)^2 8n+6=4j^2 +4j+4k^2 +4l^2 4n+3_(odd) =2(j^2 +j+k^2 +l^2 )_(even) impossible (2) 8n+7=(2j+1)^2 +(2k+1)^2 +(2l+1)^2 8n+4=4(j^2 +j+k^2 +k+l^2 +l) 2n+1_(odd) =j(j+1)_(even) +k(k+1)_(even) +l(l+1)_(even) impossible$
	$8 n + 7 = u^{2} + v^{2} + w^{2}$ $8 n + 7 is odd \Rightarrow {\begin{cases} u = 2 j + 1 \land v = 2 k \land w = 2 l \\ u = 2 j + 1 \land v = 2 k + 1 \land w = 2 k + 1 \end{cases}$ $(1)$ $8 n + 7 = {(2 j + 1)}^{2} + {(2 k)}^{2} + {(2 l)}^{2}$ $8 n + 6 = 4 j^{2} + 4 j + 4 k^{2} + 4 l^{2}$ $\underset{odd}{\underset{⏟}{4 n + 3}} = \underset{even}{\underset{⏟}{2 (j^{2} + j + k^{2} + l^{2})}}$ $impossible$ $(2)$ $8 n + 7 = {(2 j + 1)}^{2} + {(2 k + 1)}^{2} + {(2 l + 1)}^{2}$ $8 n + 4 = 4 (j^{2} + j + k^{2} + k + l^{2} + l)$ $\underset{odd}{\underset{⏟}{2 n + 1}} = \underset{even}{\underset{⏟}{j (j + 1)}} + \underset{even}{\underset{⏟}{k (k + 1)}} + \underset{even}{\underset{⏟}{l (l + 1)}}$ $impossible$
	Commented by MrGaster last updated on 18/Oct/24

	Excluding these 8n+7, there are still four numbers left, and I find that they are related to 8n+7: they can be written in the form of 4 a * (8n+7). Please prove that all 4 a* (8n+7) cannot be written in the form of the sum of at least three squares.
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