Solving Integer Cubes & Squares: 7k & 3k?

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I am trying to solve how an integer is simultaneously that is simultaneously a square and a cube number must be either of the form 7k or 7k+1 and I am failing when i work (7k+2)^2, even (7k+3)^2....Can i interpret 7k+1 as 7k+(-1), i think i can't but then i fail in many steps! Also i know that any square number can be expressed as 3k or 3k+1...IS THERE SOMETHING SIMILAR FOR THE CUBIC PLEASE?
 
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MAGICMATHS said:
I am trying to solve how an integer is simultaneously that is simultaneously a square and a cube number must be either of the form 7k or 7k+1 and I am failing when i work (7k+2)^2, even (7k+3)^2....Can i interpret 7k+1 as 7k+(-1), i think i can't but then i fail in many steps! Also i know that any square number can be expressed as 3k or 3k+1...IS THERE SOMETHING SIMILAR FOR THE CUBIC PLEASE?

The easiest way is using Fermat's Little Theorem.

Let a be any integer. If a is both a square and a cube, it is a sixth power.

By Fermat's Little Theorem,

a^7 \equiv a \pmod 7

Two cases:

Case 1: 7 \nmid a

Hence, a^6 \equiv 1 \pmod 7 \Rightarrow a^6 = 7k + 1, k \in \mathbb{Z}.

Case 2: 7 | a

Hence, a^6 \equiv 0 \pmod 7 \Rightarrow a^6 = 7k, k \in \mathbb{Z}.

So a^6 is either of the form 7k or 7k + 1 (QED).

You can construct something exactly like this for the squares modulo 3, and indeed generalise it for any (p-1)th power modulo a prime p.

Do you think the analogous statement for cubes would be true, given that 4 is composite?
 
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I don't know Fermat`s little Theory, i just started number theory to be honest ..is there a possibility using div alg pls?
 
MAGICMATHS said:
I don't know Fermat`s little Theory, i just started number theory to be honest ..is there a possibility using div alg pls?

Well, you could use the longer method presented here: http://www.mathyards.com/vb/showthread.php?691-Division-Algorithm-help !
 
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MagicMaths, if this is homework or coursework, could you please post it in the appropriate homework forum (precalculus mathematics) in future, please?
 

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