Sum of Two Squares: Intro to Number Theory

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SUMMARY

The discussion centers on the mathematical concept of representing integers as sums of two squares, particularly highlighting Fermat's theorem which states that any prime of the form 4k+1 can be uniquely represented as such. Additionally, Lagrange's theorem asserts that any positive integer can be expressed as the sum of four squares. The conversation emphasizes the theoretical nature of this topic, suggesting that while there may not be practical applications, the study of number theory is valuable for its intrinsic interest.

PREREQUISITES
  • Understanding of Fermat's theorem and its implications
  • Familiarity with Lagrange's theorem in number theory
  • Basic knowledge of the sum of squares function, denoted as r2(n)
  • Interest in theoretical mathematics and number theory concepts
NEXT STEPS
  • Research "representations of integers as sums of squares"
  • Study Fermat's theorem and its applications in number theory
  • Explore Lagrange's theorem and its significance in mathematical proofs
  • Investigate the properties and applications of the sum of squares function r2(n)
USEFUL FOR

Students of mathematics, particularly those interested in number theory, educators teaching introductory courses, and anyone seeking to deepen their understanding of mathematical theorems related to integer representations.

matqkks
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Why bother writing a given integer as the sum of two squares? Does this have any practical application? Is there an introduction on a first year number theory course which would motivate students to study the conversion of a given integer to sums of two squares?
 
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There is a good deal of material. For instance the corollary to a theorem of Fermat :- "Any prime ##p## of the form ##4k+1## can be represented uniquely as the sum of 2 squares. Or (related) Lagrange's theorem:- "Any positive integer ##n## can be written as the sum of 4 squares, some of which may be 0".
Then there are also many interesting properties of ##r_2(n)## where ##r## is the sum of squares function.
I would recommend you research "representations of integers as sums of squares."
 
matqkks said:
Why bother writing a given integer as the sum of two squares? Does this have any practical application?

It most likely does not. There are applications of number theory, but overall you should take the class mainly because you find it interesting, not because of possible applications.
 

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