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#1 2014-04-27 22:17:42
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'
I'm not crazy, my mother had me tested.
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#2 2014-04-27 22:28:36
- Bob
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- Registered: 2010-06-20
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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
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
Sometimes I deliberately make mistakes, just to test you! …………….Bob 
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#3 2014-04-27 22:33:51
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'
I'm not crazy, my mother had me tested.
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#4 2014-04-27 22:46:18
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Number Theory - Arithmetic Progression
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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#5 2014-04-27 22:51:13
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'
I'm not crazy, my mother had me tested.
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#6 2014-04-27 22:57:47
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
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.
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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#7 2014-04-27 22:59:30
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'
I'm not crazy, my mother had me tested.
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#8 2014-04-27 23:02:40
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
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.
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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#9 2014-04-27 23:04:50
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'
I'm not crazy, my mother had me tested.
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#10 2014-04-27 23:11:12
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
Re: Number Theory - Arithmetic Progression
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.
Offline
#11 2014-04-27 23:12:47
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'
I'm not crazy, my mother had me tested.
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#12 2014-04-27 23:15:42
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Offline
#13 2014-04-27 23:17:14
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'
I'm not crazy, my mother had me tested.
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#14 2014-04-27 23:18:45
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Offline
#15 2014-04-27 23:22:11
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'
I'm not crazy, my mother had me tested.
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#16 2014-04-27 23:25:29
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Offline
#17 2014-04-27 23:29:07
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'
I'm not crazy, my mother had me tested.
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#18 2014-04-27 23:33:16
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
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.
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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#19 2014-04-27 23:37:08
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'
I'm not crazy, my mother had me tested.
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#20 2014-04-27 23:43:43
- bobbym
- bumpkin
- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606
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.
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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#21 2014-04-28 00:02:45
- Bob
- Administrator
- Registered: 2010-06-20
- Posts: 10,643
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 (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
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#22 2014-04-28 00:18:29
Re: Number Theory - Arithmetic Progression
hi bob;
Thank You 
'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.
Offline
#23 2014-04-28 22:44:13
- Bob
- Administrator
- Registered: 2010-06-20
- Posts: 10,643
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 (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
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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#24 2014-04-29 00:42:43
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'
I'm not crazy, my mother had me tested.
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#25 2014-04-29 19:55:19
- Bob
- Administrator
- Registered: 2010-06-20
- Posts: 10,643
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
Last edited by Bob (2014-04-29 20:05:02)
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
Sometimes I deliberately make mistakes, just to test you! …………….Bob
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