Understand the problem

 Let \(a,b,c \in \mathbb{N}\) be such that
\(a^2+b^2=c^2\) and \(c-b=1\).
Prove that
(i) \(a\) is odd,
(ii) \(b\) is divisible by 4,
(iii) \(a^b+b^a\) is divisible by \(c\).  

Source of the problem
I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2018. Subjective Problem no. 7.
Topic
Number Theory

Difficulty Level
8 out of 10

Suggested Book

Elementary Number Theory‘ by David M. Burton
‘Challenge and Thrill of Pre-College Mathematics’ by V,Krishnamurthy, C.R.Pranesachar, ect.

Start with hints

Do you really need a hint? Try it first!

\(a^2+b^2=c^2=(b+1)^2=b^2+2b+1\) \(\Rightarrow a^2=2b+1\) Which implies \(a^2\) is odd integer. \(\Rightarrow a\) is also an odd integer =\(2k+1\) (say).

\((2k+1)^2+b^2=c^2\) \(\Rightarrow 4k^2+4k+1+b^2=b^2+2b+1\) \(\Rightarrow b=2k(k+1)\). Now \(k(k+1) \) is always even =\(2l\) (say). Therefore, \(b=4l\), i.e. \(b\) is divisible by 4.  

\(a^b+b^a=a^{4l}+b^{2k+1}=(a^2)^{2l}+(c-1)^{2k+1}\) =\((2b+1)^{2l}+(c-1)^{2k+1}\) =\((2c-1)^{2l}+(c-1)^{2k+1}.\)

Now \(2l\) is even , therefore , \((2c-1)^{2l} \) is of the form : \( 2cp+1\) where \(p \in \mathbb{N}\). And \(2k+1\) is odd , therefore \((c-1)^{2k+1} \) is of the form : \(cq-1\), where \(q \in \mathbb{N}\). Therefore \(a^b+b^a=(2cp+1)+(cq-1)=c\cdot (2p+q)\). \(\Rightarrow c|a^b+b^a\).

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