Proof by induction of the sum of 2 squares

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SUMMARY

The discussion focuses on proving by induction that any integer \( r = a_1 a_2 \ldots a_n \), where each \( a_i \) is a sum of two squares, is itself a sum of two squares. The algebraic identity used is \( (a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2 \). The base case starts with \( n=2 \), and the proof proceeds by assuming the statement holds for \( n=k \) and showing it for \( n=k+1 \). Examples provided include integers such as 2, 5, and 8, which can be expressed as sums of two squares.

PREREQUISITES
  • Understanding of mathematical induction
  • Familiarity with algebraic identities, specifically \( (a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2 \)
  • Knowledge of sums of squares in number theory
  • Basic experience with proof writing in mathematics
NEXT STEPS
  • Study the principles of mathematical induction in depth
  • Explore the properties of sums of squares in number theory
  • Practice proving algebraic identities similar to \( (a^2 + b^2)(c^2 + d^2) \)
  • Investigate examples of integers expressed as sums of two squares
USEFUL FOR

Mathematics students, particularly those studying number theory or proof techniques, educators teaching mathematical induction, and anyone interested in algebraic identities and their applications in proofs.

  • #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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