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#1 2014-04-27 22:17:42

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
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Number Theory - Arithmetic Progression

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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#2 2014-04-27 22:28:36

bob bundy
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Registered: 2010-06-20
Posts: 6,426

Re: Number Theory - Arithmetic Progression

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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#3 2014-04-27 22:33:51

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
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Re: Number Theory - Arithmetic Progression

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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#4 2014-04-27 22:46:18

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Number Theory - Arithmetic Progression


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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#5 2014-04-27 22:51:13

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
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Re: Number Theory - Arithmetic Progression

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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#6 2014-04-27 22:57:47

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Number Theory - Arithmetic Progression

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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#7 2014-04-27 22:59:30

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
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Re: Number Theory - Arithmetic Progression

Sorry, my computer has an OS problem.

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


'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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#8 2014-04-27 23:02:40

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Number Theory - Arithmetic Progression

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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#9 2014-04-27 23:04:50

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
Website

Re: Number Theory - Arithmetic Progression

I have got two results so far:
25 divides c
3 divides 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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#10 2014-04-27 23:11:12

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Number Theory - Arithmetic Progression

That means

and


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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#11 2014-04-27 23:12:47

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
Website

Re: Number Theory - Arithmetic Progression

Its c, not x

Which means

75 divides c

But I do not know what to do next


'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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#12 2014-04-27 23:15:42

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Number Theory - Arithmetic Progression

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

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#13 2014-04-27 23:17:14

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
Website

Re: Number Theory - Arithmetic Progression

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


'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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#14 2014-04-27 23:18:45

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Number Theory - Arithmetic Progression

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

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#15 2014-04-27 23:22:11

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
Website

Re: Number Theory - Arithmetic Progression

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


'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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#16 2014-04-27 23:25:29

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Number Theory - Arithmetic Progression

You could just plug them in with a calculator.


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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#17 2014-04-27 23:29:07

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
Website

Re: Number Theory - Arithmetic Progression

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


'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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#18 2014-04-27 23:33:16

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Number Theory - Arithmetic Progression

Let me see if I can get one more equation.


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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#19 2014-04-27 23:37:08

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
Website

Re: Number Theory - Arithmetic Progression

Try bob's


'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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#20 2014-04-27 23:43:43

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,237

Re: Number Theory - Arithmetic Progression

There are too many variables and equations.


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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#21 2014-04-28 00:02:45

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,426

Re: Number Theory - Arithmetic Progression

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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#22 2014-04-28 00:18:29

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
Website

Re: Number Theory - Arithmetic Progression

hi bob;

Thank You big_smile


'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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#23 2014-04-28 22:44:13

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,426

Re: Number Theory - Arithmetic Progression

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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#24 2014-04-29 00:42:43

Agnishom
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From: The Complex Plane
Registered: 2011-01-29
Posts: 17,172
Website

Re: Number Theory - Arithmetic Progression

How can I get them? Brute force?


'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'
'You have made another human being happy. There is no greater accomplishment.' -bobbym

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#25 2014-04-29 19:55:19

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,426

Re: Number Theory - Arithmetic Progression

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

View Image: aggi.gif

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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