Can Any Integer Congruent to 3 Modulo 4 Be Expressed as a Sum of Two Squares?

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SUMMARY

The discussion centers on proving that any integer \( a \) congruent to 3 modulo 4 cannot be expressed as the sum of two squares, specifically \( a \neq c^2 + d^2 \) for any integers \( c \) and \( d \). The participant demonstrates that integers of the form \( 4k + 3 \) do not yield valid pairs \( (c, d) \) that satisfy the equation. The analysis includes testing various values and highlights the need to explore the properties of squares modulo 4 to solidify the proof.

PREREQUISITES
  • Understanding of modular arithmetic, specifically modulo 4.
  • Familiarity with integer properties and representations.
  • Basic knowledge of quadratic residues.
  • Experience with proofs in number theory.
NEXT STEPS
  • Study the properties of quadratic residues modulo 4.
  • Explore the proof of Fermat's theorem on sums of two squares.
  • Investigate the implications of integers in different congruence classes.
  • Learn about the structure of integers in the context of modular arithmetic.
USEFUL FOR

Mathematics students, particularly those studying number theory, educators teaching modular arithmetic, and anyone interested in the properties of integers and quadratic forms.

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Homework Statement


All work is done in the set of integers, Z
a is congruent to 3 modulo 4, Want to prove a=/= c^2 + d^2 for any c or d

Homework Equations


The Attempt at a Solution


My work:
a congruent to 3 modulo 4 => a -3 = 4k
or a = 4k +3
Then want to prove that: 4k+3 =/= c^2 + d^2 *** for any c or dI've tested out some experimental values for 4k+3 and indeed none of them have any c or d that match this equation but I'm not sure how to progress from *** onwards, It is probably a very simple matter but I can't wrap my head around this.
 
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pandaBee said:

Homework Statement


All work is done in the set of integers, Z
a is congruent to 3 modulo 4, Want to prove a=/= c^2 + d^2 for any c or d

Homework Equations





The Attempt at a Solution


My work:
a congruent to 3 modulo 4 => a -3 = 4k
or a = 4k +3
Then want to prove that: 4k+3 =/= c^2 + d^2 *** for any c or d


I've tested out some experimental values for 4k+3 and indeed none of them have any c or d that match this equation but I'm not sure how to progress from *** onwards, It is probably a very simple matter but I can't wrap my head around this.

You can express ANY integer c as 4k, 4k+1, 4k+2 or 4k+3. What are the possible expressions for c^2? In other words, what are the possible values of c^2 mod 4?
 

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