Lagrange's Four-Square Theorem: 8n-1 Sum of 4 Squares?

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Discussion Overview

The discussion revolves around Lagrange's four-square theorem and its application to the expression of numbers of the form 8n-1 as sums of four squares. Participants explore whether all such numbers can be expressed in this way and seek proofs or explanations for their observations.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant states that the first few values of 8n-1 can only be expressed as the sum of four squares, questioning if this holds for all n.
  • Another participant mentions the possible values of squares modulo 8 being 0, 1, and 4, suggesting that only one combination can yield 7 using a maximum of four numbers.
  • Several examples are provided, showing specific representations of numbers like 7, 15, 23, and 31 as sums of four squares.
  • There is a request for clarification on the concept of modulo 8 and its relevance to the discussion.
  • One participant attempts to explain modular arithmetic using a clock analogy, relating it to the sums of squares and their modular properties.
  • A participant recalls a previous encounter with the topic in a math test, indicating some familiarity but also a need for further clarification.

Areas of Agreement / Disagreement

Participants express differing levels of understanding regarding the application of modular arithmetic to the problem. While some agree on the properties of squares modulo 8, the overall question of whether all values of 8n-1 can be expressed as sums of four squares remains unresolved.

Contextual Notes

Some participants express uncertainty about modular arithmetic and its implications for the discussion, indicating a potential limitation in understanding the mathematical framework being discussed.

karpmage
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Lagrange's four-square theorem states that any natural number can be expressed as the sum of four integer squares. I've noticed that the first few values of 8n-1 can all only be expressed as a minimum of the sum of four squares. Is this true for all values of n? What's the proof behind it? Thanks.
 
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karpmage said:
Lagrange's four-square theorem states that any natural number can be expressed as the sum of four integer squares. I've noticed that the first few values of 8n-1 can all only be expressed as a minimum of the sum of four squares. Is this true for all values of n? What's the proof behind it? Thanks.

The possible values of squares modulo 8 are 0, 1 and 4. There's only one way to make 7 from 0, 1 and 4 using a maximum of 4 numbers.
 
By what i said i mean

7 = 2^2 + 1^2 + 1^2 + 1^2
15 = 3^2 + 2^2 + 1^2 + 1^2
23 = 3^2 + 3^2 + 2^2 + 1^2
31 = 3^2 + 3^2 + 3^2 + 2^2
etc.

i.e. The first few values of 8n-1 (The first 10 at least) can't be shown as a sum of less than four squares.

Not sure what you mean by modulo 8 and what-not, sorry. You might have to dumb it down a bit for me. What exactly are you trying to say?
 
willem2 said:
The possible values of squares modulo 8 are 0, 1 and 4. There's only one way to make 7 from 0, 1 and 4 using a maximum of 4 numbers.

Sorry, meant to quote you in previous post.
 
Think about clock work math. If it's 11 a.m. and add two more hours we get 1 p.m. You can view this as 11 mod 12 + 2 mod 12 = 1 mod 12. So what he's saying is that the values in mod 8 that are squared can only be 0, 1, and 4. So now if you consider 7 mod 8, you can only get that by adding 1 + 1 + 1 + 4. If you're not familiar with this method:

http://en.wikipedia.org/wiki/Modula...3osI9mtGGTpQGAAAABAEAADqRgAAgHAAAAAAAALgaAAA=
 
MarneMath said:
Think about clock work math. If it's 11 a.m. and add two more hours we get 1 p.m. You can view this as 11 mod 12 + 2 mod 12 = 1 mod 12. So what he's saying is that the values in mod 8 that are squared can only be 0, 1, and 4. So now if you consider 7 mod 8, you can only get that by adding 1 + 1 + 1 + 4. If you're not familiar with this method:

http://en.wikipedia.org/wiki/Modula...3osI9mtGGTpQGAAAABAEAADqRgAAgHAAAAAAAALgaAAA=

Ah, I remember now. This came up on a maths test I once did. Completely forgot about it though. Thanks for the help.
 

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