Can someone show me the proof by mathematical induction that isn't divisible by 49 for every n (natural number).

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- Aug 23rd 2006, 07:45 PMOReillyDivisible by 49
Can someone show me the proof by mathematical induction that isn't divisible by 49 for every n (natural number).

- Aug 23rd 2006, 07:55 PMThePerfectHackerQuote:

Originally Posted by**OReilly**

- Aug 23rd 2006, 08:09 PMOReillyQuote:

Originally Posted by**ThePerfectHacker**

- Aug 23rd 2006, 08:23 PMThePerfectHackerQuote:

Originally Posted by**OReilly**

I was just writing out a proof. But I accidently closed my internet thing and lost everything I typed. - Aug 23rd 2006, 11:32 PMrgep
Firstly observe that n^2+n+2 is divisible by 7 if and only if n is congruent to 3 modulo 7: that is, if n is of the form 7x+3. See this by writing it as n^2 - 6n + 9 + 7(n-1), so that it is (n-3)^2 modulo 7.

Now substitute n=7x+3 into n^2 + n + 2 to get 49x^2 + 49x + 14, which cannot be divisible by 49. - Aug 24th 2006, 07:23 AMOReilly
Is this proof valid?

Supose that for expression is divisible by 49:

Supose that now for expression is also divisible by 49:

From first equation we get that .

Substituting into second equation we get:

Last expression isn't divisible by 49 so isn't divisible by 49. - Aug 24th 2006, 03:40 PMThePerfectHacker
You need to show that,

If,

then,

Equivalently (contropositive),

If,

then,

You shown that:

if,

then,

This is not equivalent to the above two statements. - Aug 24th 2006, 06:10 PMQuickI don't get it...
I don't understand your proof.

where am I off?

If

then,

:confused: - Aug 24th 2006, 06:57 PMThePerfectHacker
Here is how you do it.

---

You need to show there doth not exist such a such as,

Which means there are no integral solution for .

If you solve for "n",

.

Note the following, if the expression in the radical is a square then "n" is an integral number because the square root of an odd square is an odd whole number and when added of subtracted to 1 produces an even number and when finally divided by two produces an integral number, thus, if

is not a square then,

is also not a square.

Thus, again you need to show that.

IF,

is not a square

THEN,

is not a square.

~~Contropositive Time~~

IF,

is a square.

THEN,

is a square.

For some reason I cannot continue. I am hoping someone will see this and tell me the next step. - Aug 24th 2006, 10:15 PMCaptainBlackQuote:

Originally Posted by**OReilly**

by for consecutive integral values of .

RonL - Aug 24th 2006, 10:35 PMCaptainBlackQuote:

Originally Posted by**Quick**

**Im**PerfectHacker agrees with you and he doesn't understand his

argument either :D

It is either so subtle that we can't see the point, or its wrong. I know

where I'm putting my money (no not in that box under the bed).

RonL - Aug 25th 2006, 05:32 AMThePerfectHacker
Look at post #7. I say it is no good.

---

I still do not see why you chose such a method. The quickest would be quadradic reciprocity. - Aug 25th 2006, 06:54 AMGlaysherQuote:

Originally Posted by**ThePerfectHacker**

Need to prove either

statement true for n = k implies statement true for n = k + 1

or

Contropositive

Statememt not true for n = k + 1 implies statement not true for n = k

OReilly showed

Statement true for n = k implies statement not true for n = k + 1 - Aug 25th 2006, 07:34 AMQuickQuote:

Originally Posted by**Glaysher**

- Aug 25th 2006, 09:56 AMGlaysherQuote:

Originally Posted by**Quick**

Usually you are trying to prove something for all natural numbers n

You show that it is true when n = 1

Then you assume it is true for some n = k

It is okay to assume this as we have already proved it for n = 1 so we already know there is at least one possible value for k.

We then try to prove that it also holds for n = k + 1 assuming that it holds for n= k.

Once we have done this we have proved it for all natural numbers n as

True for n = 1 means true for n = 2 means true for n = 3 and so on ...

This method can be adapted to suit several different purposes