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Algebra

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11 Mar 2008 20:00:10 IST

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Prove using induction or otherwise that the equation x²+y² = zⁿ has solutions in N for all n belonging to the set N of natural numbers.

i.e. for any natural number n, we can always find (x,y,z) satisfying x²+y² = zⁿ

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Hari Shankar

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11 Mar 2008 21:51:44 IST

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Why is this being ignored?

no number theory at all friends.

I mean x²+y² = z obviously has solutions,

x²+y² = z² has Pythagorean triplets as solutions.

So, can you prove that likewise x²+y² = z³, x²+y² = z⁴and so on also have got solutions in natural numbers.

sreeraman nagasubramaniyan

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12 Mar 2008 15:26:57 IST

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1)obviously x²+y² = z has solutions

2)now x²+y² = z² has pythagorean triplets as solutions

these 2 sir himself pointed out

now we can have point 2) as our base

whenever x,y,z are pythagorean triplets ...ax,ay,az are also pythagorean triplets..

now easy technique thus to find solution of

x² + y² = z⁴

would be to ..take any triplet

eg: 3 4 5

multiply all numbers commonly by the number 5

so we get

15 20 25

now here since they are also triplets

15²+20² = 25² = 5⁴

now to get it of the form

x² + y² = z⁶ ...do the same process again!

we get

75² + 100² = 125² = 5⁶

so thus we can obtain all even sequences

now for the odd ones ...

as shown above

6 8 10 form a triplet

so multiplying by 800

we get

4800 6400 8000

4800² + 6400² = 8000² = 400³

now again same as above ..a manipulation will make u represent sum of two squares as power of 5 of an integer ..and then 7 and so on ...

thus we can find solutions in N for all n belonging to n for some x y z

satisfying

x²+y² = zⁿ

Hari Shankar

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12 Mar 2008 18:17:42 IST

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In addition to sboosy's method, there are two ways of looking at it, both of which give us some additional info:

The basis for the both is the fact that whenever a number z can be written as z = x₁²+y₁², we can find x₂, y₂ such that x₂²+y₂² = z² (Notice it is the same z)

This is because z² = ( x₁²+y₁²)² = (x₁²-y₁²)²+ (2x₁y₁)²

1. Now if we have x_n²+y_n² = zⁿ, then multiplying by z² gives

(x_nz)²+(y_nz)² = zⁿ⁺².

This can be used to prove the hypothesis for all even numbers by starting from n=2 and for all odd numbers starting from n=1.

2. Otherwise use the identity (x₁²+y₁²) (x_n²+y_n²) = (x₁x_n-y₁y_n)² + (x₁y_n+x_ny₁)²

So now you can prove the hypothesis using induction.

Nadeem

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12 Mar 2008 18:25:47 IST

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Number theory is ignored because it's not in the JEE syllabus. otherwise if all the jee- aspirants had learned number theory , then india would have got a lot more medals in the IMO

Hari Shankar

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12 Mar 2008 18:36:49 IST

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Some time ago, I was going thru a book that had compiled maths olympiads from various countries, NMOs, RMOs etc. I was embarassed to see that India never figures in any of these. China, USA, Vietnam(!), the Eastern European countries(Romania, Bulgaria, Hungary), these are the ones that dominate, and win medals at the IMO. In fact, if you see John Scholes site (that kalva demon thing), some of the qns at our INMOs are shown to be wrong or incomplete

Vietnam has a forum dedicated only to solving inequalities. It is just mind-blowing stuff because this is one area where considerable ingenuity is required.

We have a million miles to go.

venkatraman gurumurthy

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13 Mar 2008 11:34:53 IST

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