Puzzling Puzzle

cheer99 · 2005-05-19 17:29:00

Hi All-

I need some help. I'm so bad at math.

Here is the questions:

what is the number of prime numbers between one and a million that become a perfect square if you add 1 to them?

Thanks a bunch!
Delina

MathsIsFun · 2005-05-19 17:57:40

You must be good at math to be even attempting that one!

By "perfect square" I will assume that you mean the squares of the whole numbers 1,2,3,4 ... which are 1,4,9,16 ... etc

So, to start with, you are dealing with 1000 candidate numbers, because 1000^2 = 1,000,000 and your question said that is as far as you need go.

You could use a spreadsheet to generate those 1,000 numbers, subtract 1 in the next column and compare that to a list of primes. That would give you your answer.

There may be a way of cutting the problem down a little bit by using the properties of primes or perfect squares ...

Let me see, a perfect square is n^2 and you are asked to see if that is prime when you subtract one, or:

Is n^2-1 a prime (for n=1,2,3,...1000)

Now, a simple property of primes is that they are odd numbers (except for 2), so that would mean that the perfect squares must be even (because you subtract one). So that eliminates half of the perfect squares right there (even times even is even, odd times odd is odd)

So, we are now trying to check the squares of 2,4,8, ... 1000

For example:

2*2 = 4, -1 = 3, is 3 prime? YES
4*4 = 16, -1 = 15, is 15 prime? NO
6*6 = 36, -1 = 35, is 35 prime? NO

Only 497 more numbers to check.

Now, I have a suspicion that there are no more "YES" answers ... because the factors work out neatly as:

2*2 = 4, -1=3 ==> has factors 1 and 3
4*4 = 16, -1=15 ==> has factors 3 and 5
6*6 = 36, -1=35 ==> has factors 5 and 7
8*8 = 64, -1=63 ==> has factors 7 and 9
10*10 = 100, -1=99 ==> has factors 9 and 11

There is a pattern!

And the pattern is this:

n^2 = (n-1)(n+1) + 1

(because (n-1)(n+1) = n^2 +n - n - 1 = n^2 - 1)

So that means NONE of them will be prime (except for 2*2-1=3), as they have factors when you subtract one !!

A lovely trick question, thanks.

cheer99 · 2005-05-20 02:56:23

thanks for the helps, but in your post you put subtract, but in my question it says add one. So is your post correct to help me?

Is the answer 1?

Thanks again for the help!

MathsIsFun · 2005-05-20 10:23:00

"prime numbers ... that become a perfect square if you add 1 "

I just turned it around to be

"perfect squares ... that become a prime number if you subtract 1 "

Yep, I believe there are just 1. But that's just me. I do get things wrong.

Roraborealis · 2005-05-20 22:03:51

Ah, so do us all.

Zach · 2005-05-21 01:43:30

I don't.

cheer99 · 2005-05-21 01:57:36

Thanks a bunch! I really appreciate it. We get an extra 10 points on the final exam if we get it right! College Math is so much harder then high school math!

Zach · 2005-05-21 01:58:43

That's probably the idea of it being a higher level math.

MathsIsFun · 2005-05-21 10:44:41

Har-de-har ...

'Tis true though, at a certain level math goes into overdrive.

Zach · 2005-05-21 21:01:07

Foolish maths. Always trying to be clever.

Roraborealis · 2005-05-22 00:17:42

Isn't that the point of school?

Vulcan · 2005-05-22 00:30:18

um............
no?
or.............
yes?
wait..........
i know this one...........................................................................
um....................
i think................
yes?

Roraborealis · 2005-05-22 00:30:58

Correct.

Vulcan · 2005-05-22 00:33:30

wahoooooooooooooooooooooo!!!!!!!!!!!!!11
i am the smartest kid in the universe!
or........
atleast the mathsisfunniverse!

Roraborealis · 2005-05-22 00:49:18

If you say so.........

Zach · 2005-05-22 04:23:51

Riiiiiight.

insomnia · 2005-05-22 20:09:26

Yes, ok. If you want to believe that fine with us. Who are we to destroy your fragile pysche.

ppl · 2005-05-22 23:39:06

Help me im stuck on the gcse topic and i really want to be a gr8 mathmatician. its so hard for me because im have a huge reputation as a i math brain but i just can do divison plzzzzzzz help me!!!!!!!!!!!!

ppl · 2005-05-22 23:40:07

ppl wrote:

Help me im stuck on the gcse topic and i really want to be a gr8 mathmatician. its so hard for me because im have a huge reputation as a i math brain but i just can do divison plzzzzzzz help me!!!!!!!!!!!!

sorry i ment CANT and i aint no novice im goin to recive the highest award in my maths class a ************8 see beat ne if u can moo ha ha ha

ppl · 2005-05-22 23:41:09

ppl wrote:

ppl wrote:
Help me im stuck on the gcse topic and i really want to be a gr8 mathmatician. its so hard for me because im have a huge reputation as a i math brain but i just can do divison plzzzzzzz help me!!!!!!!!!!!!
sorry i ment CANT and i aint no novice im goin to recive the highest award in my maths class a ************8 see beat ne if u can moo ha ha ha

fine then dont help me u r all evil...... i just goin to go in my corner and cry!! sob sob sniff

MathsIsFun · 2005-05-23 00:50:18

Hello "ppl"

There was only 2 minutes between your first and last post ... this is a forum, not a phone conversation

When you post a question here you need to give people a chance to come on the forum, read your message and reply. This can be many hours to days sometimes, but we will do the best we can to help.

So, can you tell us what your problem is with division - is it long division?

Have you tried reading our pages on it starting here

And you can practice with these [url=]worksheets[/url] - there are normal division and long division worksheets, and you can view them with or without answers.

a mad physicist · 2005-05-24 11:04:06

neopets?

MathsIsFun · 2005-05-24 18:30:31

Like this? No, sorry.

Ellie · 2005-05-25 09:03:02

cheer99 wrote:

Thanks a bunch! I really appreciate it. We get an extra 10 points on the final exam if we get it right! College Math is so much harder then high school math!

lol! More like your trying to get the ansewer for this weeks Lenny Conundrem on neopets.com!

MathsIsFun · 2005-05-25 09:41:42

Is that right? LOL !!

Do I get a prize for answering it, then?

Math Is Fun Forum

#1 2005-05-19 17:29:00

Puzzling Puzzle

#2 2005-05-19 17:57:40

Re: Puzzling Puzzle

#3 2005-05-20 02:56:23

Re: Puzzling Puzzle

#4 2005-05-20 10:23:00

Re: Puzzling Puzzle

#5 2005-05-20 22:03:51

Re: Puzzling Puzzle

#6 2005-05-21 01:43:30

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#7 2005-05-21 01:57:36

Re: Puzzling Puzzle

#8 2005-05-21 01:58:43

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#9 2005-05-21 10:44:41

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#10 2005-05-21 21:01:07

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#11 2005-05-22 00:17:42

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#12 2005-05-22 00:30:18

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#13 2005-05-22 00:30:58

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#14 2005-05-22 00:33:30

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#15 2005-05-22 00:49:18

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#16 2005-05-22 04:23:51

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#17 2005-05-22 20:09:26

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#18 2005-05-22 23:39:06

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#19 2005-05-22 23:40:07

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#20 2005-05-22 23:41:09

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#21 2005-05-23 00:50:18

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#22 2005-05-24 11:04:06

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#23 2005-05-24 18:30:31

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#24 2005-05-25 09:03:02

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#25 2005-05-25 09:41:42

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