Proof by induction of the sum of 2 squares

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Homework Help Overview

The discussion revolves around proving by induction that any integer expressed as a product of sums of two squares is itself a sum of two squares. The original poster presents an algebraic identity and examples to illustrate the concept, seeking clarification on how to set up the proof.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the interpretation of the notation used in the problem, particularly the meaning of r=a1,a2,a3...an. There are attempts to clarify the relationship between the integers involved and the algebraic identity provided.

Discussion Status

Several participants express confusion about the problem statement and the setup for the proof. Some guidance is offered regarding the algebraic identity and the steps needed to prove the statement for n=2, followed by induction for n=k and n=k+1.

Contextual Notes

Participants note that the problem requires understanding the relationship between the sums of squares and the algebraic identity, as well as the implications of the induction process. There is mention of a JPEG attachment containing the problem statement, which may provide additional context.

  • #31
so I've gone along with

assume: rk=(a2+b2)
if, rk+1=(a2+b2)(c2+d2)=(ac-bd)2)+(ad+bc)2
 
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  • #32
aporter1 said:
so I've gone along with

assume: rk=(a2+b2)
if, rk+1=(a2+b2)(c2+d2)

Then rk+1=(ac-bd)2)+(ad+bc)2
Is more like it.
 
  • #33
so I'm on the right track for the right side then?
 
  • #34
aporter1 said:
so I'm on the right track for the right side then?
What do you mean by "the right side" ?
 
  • #35
(ac−bd)2+(ad+bc)2
 
  • #36
Mod note: Moved this thread to Precalculus section.
 
  • #37
aporter1 said:
so I've gone along with

assume: rk=(a2+b2)
if, rk+1=(a2+b2)(c2+d2)=(ac-bd)2)+(ad+bc)2
Let me answer this post of yours again.

It makes sense to usea subscript with r. Did your teacher give you that idea?

You should really define rn somewhere.

Like: Let rn = a1a2a3a4...an where each ai is the sum of two integers.

Now, let's redo what you have above with some changes that I will put in RED.
aporter1 said:
so I've gone along with

Assume: rk=(a2+b2), for two integers, a and b.

If ak+1 = c2+d2, for two integers, c and d,

[STRIKE]if,[/STRIKE] then rk+1=(a2+b2)(c2+d2)=(ac-bd)2)+(ad+bc)2

Therefore, rk+1 is the sum of the squares of two integers.
 
  • #38
aporter1 said:
I've taken history of math discrete math
Is that two different classes or one? If it's one class, what are the prerequisites for the class you're currently in, and if there are any, did you take them and get a reasonably good grade? I get the sense that you're way over your head in this class.
aporter1 said:
, but its been a while. Its a 400 level course, its an independent study class. I've gone to my teacher for help but I don't understand
 
  • #39
Mark44 said:
Is that two different classes or one? If it's one class, what are the prerequisites for the class you're currently in, and if there are any, did you take them and get a reasonably good grade? I get the sense that you're way over your head in this class.

its one class, and i got good grades in the pre requisites. but see its an independent study class where its a special topics class, so my teacher just randomly picked a book
 
  • #40
aporter1 said:
its one class, and i got good grades in the pre requisites. but see its an independent study class where its a special topics class, so my teacher just randomly picked a book
I doubt that it was a random pick !
 

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