Exploring the Pattern of Prime Numbers and Squares

In summary, the conversation is about a pattern involving prime numbers and their squares minus one. The pattern shows that starting with the third p^2 - 1, all subsequent p^2 - 1 can be rewritten as an earlier p^2-1 times a prime number or multiple prime numbers. The person asking the question wonders if this is interesting or if they are missing something. The other person responds by saying it is a trivial observation and that it is likely just a property of small numbers. They also mention that when looking at larger numbers, this pattern is not likely to hold true.
  • #1
nate808
542
0
I am curious as to whether this pattern will always hold true:

Let's say we take the prime numbers:
2,3,5,7,11,13,17,19,23...primes
and we take the square(individually) minus 1
3,8,24,48,120,168,288,360,528...p^2 - 1

Then starting with the third p^2 - 1 (24), all of the p^2 - 1 can be rewritten as and earlier p^2-1 times a prime # (or multiple prime #'s)

for example: 24=8x3,48=24x2,528=48x11,...and so on.

Is this at all interseting or just something stupid that I am missing?

Let me know,
Thanks
 
Physics news on Phys.org
  • #2
p^2-1 times a prime # (or multiple prime #'s)
Why not just say "p²-1 times a number"?


It's a rather trivial observation; 3 is on your list.
 
Last edited:
  • #3
The number 4 times out of five is a single prime #. It seemed a lot nicer when i posted it and in the first ten i checked 9 of them were single prime numbers. I have since found 3 more where the # is a multiple of 2 primes (2 and 3 in all cases). Thanks for at least looking at it Hurkyl
 
  • #4
I have since found 3 more where the # is a multiple of 2 primes (2 and 3 in all cases). Thanks for at least looking at it Hurkyl
I suspect what you're seeing is simply a property of small numbers, rather than anything special about your construction. That's why I asked how far you looked -- I imagine when you start looking at numbers in the trillions, you won't see this sort of thing very frequently at all.
 
  • #5
Since you can factor p^2-1 you should start to see that this is not likely to be a recurrent pattern (an even weaker hypothesis is for every prime p, either p-1 or p+1 to have a repeated factor). This seems highly unlikely to be true in general.
 

1. What are prime numbers and squares?

Prime numbers are numbers that are only divisible by 1 and itself. Squares are numbers that can be written as the product of two equal numbers, or the result of multiplying a number by itself.

2. How do you find prime numbers?

The most common method is to use a process called "sieve of Eratosthenes". This involves creating a list of numbers and crossing out all the multiples of each number until only prime numbers are left.

3. What is the connection between prime numbers and squares?

There is no direct connection between prime numbers and squares. However, every prime number (except 2) can be written as the sum of two squares.

4. Are there any patterns in the distribution of prime numbers and squares?

Yes, there are many patterns and relationships between prime numbers and squares that have been discovered by mathematicians. Some of these patterns have been proven, while others are still being studied.

5. Why is exploring the pattern of prime numbers and squares important?

Understanding the patterns and relationships between prime numbers and squares can help us solve complex mathematical problems and improve our understanding of number theory. It also has practical applications in fields such as cryptography and computer science.

Similar threads

  • Programming and Computer Science
Replies
22
Views
636
  • Precalculus Mathematics Homework Help
Replies
16
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
8
Views
859
  • Precalculus Mathematics Homework Help
Replies
3
Views
900
  • Linear and Abstract Algebra
Replies
2
Views
990
  • Precalculus Mathematics Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
15
Views
3K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
692
Back
Top