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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?

'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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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,390

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

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?

'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'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,569

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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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'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,569

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.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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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'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,569

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.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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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'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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That means

and

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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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'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Based on what you have you will now have to try 75, 150, 225, 300, 375, 450...

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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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'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,569

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

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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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'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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You could just plug them in with a calculator.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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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'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Let me see if I can get one more equation.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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Try bob's

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

'Humanity is still kept intact. It remains within.' -Alokananda

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,569

There are too many variables and equations.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,390

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)*

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'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,390

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)*

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'

'Humanity is still kept intact. It remains within.' -Alokananda

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,390

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)*

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