Math Is Fun Forum

NIH

Member

Registered: 2005-06-14

Posts: 33

A Diophantine problem

Find all integer solutions of a(a + 1) = b(b + 2).

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2 + 2 = 5, for large values of 2.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: A Diophantine problem

Let's try an example. How about finding b for a=1:

a(a + 1) = b(b + 2)
1(1+1) = b(b+2)
2 = b²+2b 

So, let's try see if we can find a b that solves it:

b=1: b²+2b = 1²+2 = 3
b=0: b²+2b = 0
b=-1: b²+2b = (-1)²-2 = -1
b=-2: b²+2b = (-2)²-4 = 0
b=-3: b²+2b = (-3)²-6 = 3   ... oops, went straight past it

Time for thinking cap ...

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: A Diophantine problem

Is it possible to use an odds and evens approach ?

If a is odd then a(a+1) will be odd×even = even
If a is even then a(a+1) will be even×odd = even

If b is odd then b(b+2) will be odd×odd = odd
If b is even then b(b+2) will be even×even = even

Ooops ... gotta go ...

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,395

Re: A Diophantine problem

Thanks, MathsisFun.
I'll continue from where you left.
a(a+1) is certainly even.
b(b+2) is odd if b is odd.
If b is even, b(b+2) is even, and is always a multiple of 4.
a(a+1) is a multiple of 4 only if a is multiple of 4 or it is a number of the form 4n+3.
Lets assume a is a number of the form 4n+3.
a(a+1) = (4n+3)(4n+4) = 16n² + 16n + 12n + 12 = 16n² + 28n + 12
= 4(4n²+7n+3)
Let this number be equal to b(b+2)
b(b+2) = 4(4n²+7n+3)
b² + 2b - 4(4n²+7n+3) = 0
b = [-2 ± √{4 +16(4n²+7n+3)}]/4
Let n=0, we get an irrational number as b.
Let n=1, we get an irrational number as b.
Let n=2, again, we get an irrational number as b.

Can {4 +16(4n²+7n+3)} never be a perfect square?

Is it because the last digit of {4 +16(4n²+7n+3)} is 2 or 8 and not 4 or 6?

Last edited by Jai Ganesh (2005-08-09 18:35:20)

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: A Diophantine problem

There are no squares that end in 2 or 8, so if 4 +16(4n²+7n+3) always ends in 2 or 8, then we have a proof.
Take away 4: 16(4n²+7n+3)=...8 or 4. Multiples of 16 move in cycles of 5, with the last digit being 6, 2, 8, 4, 0...
This means that 4n²+7n+3 has to always be of the form 5m+3 or 4. So how do we show that?

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Why did the vector cross the road?
It wanted to be normal.

wcy

Member

Registered: 2005-08-04

Posts: 117

Re: A Diophantine problem

is it possible to use some general method as outlined by Euclid

NIH

Member

Registered: 2005-06-14

Posts: 33

Re: A Diophantine problem

There are no squares that end in 2 or 8, so if 4 + 16(4n²+7n+3) always ends in 2 or 8, then we have a proof.

But if n = 3 or 4 (mod 5), then 4 + 16(4n²+7n+3) ends in 4.

Here's a hint: add 1 to both sides of the original equation.

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2 + 2 = 5, for large values of 2.

wcy

Member

Registered: 2005-08-04

Posts: 117

Re: A Diophantine problem

then,
RHS=b²+2b+1= (b+1)²

but what abt LHS ?

Last edited by wcy (2005-08-10 12:42:04)

NIH

Member

Registered: 2005-06-14

Posts: 33

Re: A Diophantine problem

but what abt LHS ?

Then the LHS equals a² + a + 1. 

For what values of a can this be a perfect square?

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2 + 2 = 5, for large values of 2.

wcy

Member

Registered: 2005-08-04

Posts: 117

Re: A Diophantine problem

a can only be 0

NIH

Member

Registered: 2005-06-14

Posts: 33

Re: A Diophantine problem

a can only be 0

Or -1.

If you can show why those are the only two values, the problem is solved!

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2 + 2 = 5, for large values of 2.

NIH

Member

Registered: 2005-06-14

Posts: 33

Re: A Diophantine problem

Here is a solution to this one.

a(a + 1) = b(b + 2) <=> a² + a + 1 = (b + 1)².
For positive a, a² < a² + a + 1 < (a + 1)², implying there is a perfect square between consecutive squares; contradiction.

For negative a, a(a + 1) = p(p - 1), where p = -a, so there are no solutions for p - 1 > 0; i.e., a < -1.

It is then easy to check for solutions with a = -1, 0.
These are: (a,b) = (-1,-2), (-1,0), (0,-2), (0,0).

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2 + 2 = 5, for large values of 2.