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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?

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**bob bundy****Moderator**- Registered: 2010-06-20
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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

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I'll see.

What do you do after that?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

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

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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That is the answer but how did you do it?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'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: 107,190

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.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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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'

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**bobbym****Administrator**- From: Bumpkinland
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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.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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I have got two results so far:

25 divides c

3 divides c

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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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.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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Its c, not x

Which means

75 divides c

But I do not know what to do next

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**bobbym****Administrator**- From: Bumpkinland
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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.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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

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**bobbym****Administrator**- From: Bumpkinland
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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.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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

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**bobbym****Administrator**- From: Bumpkinland
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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.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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

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**bobbym****Administrator**- From: Bumpkinland
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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.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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Try bob's

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**bobbym****Administrator**- From: Bumpkinland
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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.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,744

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'

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,744

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: 7,744

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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