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a, b, c, d, e are five natural numbers in arithmetic progression.

a + b + c + d + e is the cube of some natural number.

b + c + d is the square of some natural number.

What is the least possible value of c?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,501

hi Agnishom,

This may help:

Say b = a + k c = a + 2k etc

Show that c = ½(N^3 - M^2) where N and M are the cube and square.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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I'll see.

What do you do after that?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 102,283

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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That is the answer but how did you do it?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 102,283

Whether you do this by hand or by computer you have to form the relationships. Then we can discuss how to solve them.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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Sorry, my computer has an OS problem.

I've done what bob asked but how does it help?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 102,283

This is what I had to start.

a > 0, b > 0, c > 0, d > 0, e > 0,

b = a + h

c = a + 2 h

d = a + 3 h

e = a + 4 h

a + b + c + d + e = m^3

b + c + d = n^2

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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I have got two results so far:

25 divides c

3 divides c

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 102,283

That means

and

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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Its c, not x

Which means

75 divides c

But I do not know what to do next

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 102,283

Based on what you have you will now have to try 75, 150, 225, 300, 375, 450...

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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But my OS is broken, I can't run mathematica

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 102,283

Then you will need to get a few more relationships if you can.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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Nope, I have a better idea of doing that without a computer.

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 102,283

You could just plug them in with a calculator.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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Sorry, this sum came up in a test where no electronic gadgets were allowed

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 102,283

Let me see if I can get one more equation.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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Try bob's

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 102,283

There are too many variables and equations.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,501

I too had the equations of post 8

=> 2c = n^3 - m^2

and 8c = n^3 + m^2

=> n^3 = 5c and m^2 = 3c => c is a multiple of 25 and 3 => c is a multiple of 75

By hand c = 75 n^3 = 375 which is not a cube

c = 150 n^3 = 750 ditto

c = 225 n^3 = 1125 ditto

.................................................

.................................................

c = 675 n^3 = 3375 => n = 15 and m^2 = 2025 => m = 45

better edit:

Let c = 75p

n = cube root (5 x 75p) = 5 x cube root (3p)

m = square root (3 x 75p) = 15 root p

So p must be a square and 3p a cube.

p = 9 is obvious.

Bob

*Last edited by bob bundy (2014-04-28 00:09:23)*

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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hi bob;

Thank You

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,501

That gives the least value for c as required. But there are more:

p = 9, 6561, 4782969, 3486784401, etc

Bob

*Last edited by bob bundy (2014-04-28 22:53:58)*

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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How can I get them? Brute force?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

I'm not crazy, my mother had me tested.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,501

hi Agnishom,

I did use 'educated' brute force, but, following your question, I have just developed a 'proof'.

As p is a perfect square and 3p a perfect cube, we can conclude that p has to be a power of 3. So I used Excel to make the table of values shown below.

HERE'S ANOTHER BUG I'VE DISCOVERED. IF I UPLOAD AN IMAGE AND THEN PREVIEW MY POST THE IMAGE IS LOST. SO HERE I HAVE POSTED INSTEAD OF PREVIEWING. I'VE DISCOVERED I CAN EDIT A POST CONTAINING THE IMAGE, PREVIEWING AS MUCH AS I LIKE, WITHOUT LOSING THE IMAGE.

NOW ON WITH THE POST:

So I could see that every 6th line was a solution. Can I prove this will happen?

All letters are positive integers:

let

r must have two more factors of 3, in order to be a cube so

So q squared must have a factor of 9 so write

This will happen every 6th power.

Bob

*Last edited by bob bundy (2014-04-29 20:05:02)*

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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