Show logical notation for being prime

In summary, a prime number is a positive integer with exactly two factors, 1 and itself. To represent a prime number using logical notation, one can use statements such as "n is a prime number if and only if its only divisors are 1 and itself." The logical notation for a number being prime is "n is a prime number" or "n is prime." An example of logical notation for a prime number is "5 is prime" or "if n is a prime number, then n = 2k + 1 for some integer k." To prove that a number is prime using logical notation, one can use a direct proof or a proof by contradiction.
  • #1
Painguy
120
0

Homework Statement


Show logical notation for being prime given N=(P1, P2,...Pn) +1

Homework Equations


The Attempt at a Solution


I came up with the following, but I am not sure if it makes sense (I used trial division)
([itex]\exists[/itex]x=((Pn+1)/((M>1)[itex]\wedge[/itex](M[itex]\leq[/itex]√(Pn+1))))[itex]\in[/itex][itex]N[/itex]) => (Pn+1 [itex]\neg[/itex]=Prime number)

If there exists a number Pn+1 divided by a number M greater than 1 and less than the squareroot of Pn plus 1 contained in a set of integers then Pn +1 is not a prime number.
 
Physics news on Phys.org
  • #2
The condition for [itex]P[/itex] being prime is that its only integer divisors are 1 and [itex]P[/itex]. This is [tex]
\forall n \in \mathbb{N} : n \perp P \Rightarrow n \in \{1, P\}.[/tex]
 

1. What is the definition of a prime number?

A prime number is a positive integer that is only divisible by 1 and itself. This means it has exactly two factors.

2. How do you represent a prime number using logical notation?

To represent a prime number using logical notation, we can use the statement "n is a prime number if and only if its only divisors are 1 and itself."

3. What is the logical notation for a number being prime?

The logical notation for a number being prime is "n is a prime number" or "n is prime."

4. Can you provide an example of logical notation for a prime number?

Yes, an example of logical notation for a prime number is "5 is prime" or "if n is a prime number, then n = 2k + 1 for some integer k."

5. How do you prove that a number is prime using logical notation?

To prove that a number is prime using logical notation, we can use a direct proof by showing that it has exactly two divisors, 1 and itself. We can also use a proof by contradiction by assuming that the number has more than two divisors and showing that this leads to a contradiction.

Similar threads

Bounded non-decreasing sequence is convergent
    • Calculus and Beyond Homework Help
Replies
5
Views
1K
Determine whether the integer 701 is prime by testing?
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
[Abstract Algebra] GCD and Relatively Prime Proof
    • Calculus and Beyond Homework Help
Replies
5
Views
1K
If p##\geq##5 is a prime number, show that p^{2}+2 is composite?
    • Calculus and Beyond Homework Help
Replies
3
Views
682
Trouble understanding proof- infinitely many primes?
    • Calculus and Beyond Homework Help
Replies
4
Views
3K
Difficult number theory problem proofs
    • Calculus and Beyond Homework Help
Replies
3
Views
2K
Prove that if p > 3 is a prime number, then p^2 = [....]
    • Calculus and Beyond Homework Help
Replies
16
Views
2K
Relatively prime proof involving a^n and b^n
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Prove there are infinitely many primes using Mersenne Primes
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
For ## n>1 ##, show that every prime divisor of ## n+1 ## is an odd integer
    • Precalculus Mathematics Homework Help
Replies
9
Views
1K

Hot Threads

Recent Insights

Back
Top