- #1
rrronny
- 7
- 0
Let [tex]\mathbb{P}[/tex] the set of primes. Let's [tex]p,q \in \mathbb{P}[/tex] and [tex]p \le q.[/tex] Find the pairs [tex](p,q)[/tex] such that [tex]2^p+3^q[/tex] and [tex]2^q+3^p[/tex] are simultaneously primes.
Which Pairs (p,q) Satisfy 2^p+3^q and 2^q+3^p Being Prime?
- Thread starter rrronny
- Start date
In summary, "On the primality of 2^p+3^q" is a highly studied and debated topic in mathematics that explores the properties and patterns of numbers in the form of 2^p+3^q, where p and q are prime numbers. The formula 2^p+3^q is a prime-generating polynomial that has been extensively studied to understand its properties and potential patterns in the generation of prime numbers. As of now, there is no known general rule or pattern for determining the primality of 2^p+3^q, but there have been some significant discoveries and conjectures about its behavior. Understanding the primality of 2^p+3^q has potential applications in
- #2
Borek
Mentor
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You have to show your attempts to receive help. This is a forum policy.
- #3
rrronny
- 7
- 0
Hi Borek,Borek said:
I do not have a solution for this problem...
I found only four solutions: [tex](1,1), (1,2), (2,3), (2,6).[/tex]
- #4
quantumdoodle
- 17
- 0
wait... 6 is not a prime...
- #5
ramsey2879
- 841
- 3
quantumdoodle said:
Neither is 1 so only one of the 4 pairs posted is acceptable. There is still another very obvious pair that was overlooked.
- #6
rrronny
- 7
- 0
Sorry...quantumdoodle said:
I meant the sixth prime number. In summary, then, the only solutions [tex](p,q)[/tex] that I found are [tex](2,2),(2,3),(3,5), (3,13).[/tex]
1. What is the significance of "On the primality of 2^p+3^q" in mathematics?
"On the primality of 2^p+3^q" is a highly studied and debated topic in mathematics, specifically in the field of number theory. It explores the properties and patterns of numbers in the form of 2^p+3^q, where p and q are prime numbers. This topic is important because it has implications for understanding the distribution of prime numbers and their relationship with other mathematical concepts.
2. How does the formula 2^p+3^q relate to the concept of primality?
The formula 2^p+3^q is a form of a prime-generating polynomial, meaning that it generates prime numbers for certain values of p and q. This formula has been extensively studied by mathematicians to understand its properties and potential patterns in the generation of prime numbers.
3. Is there a general rule or pattern for determining the primality of 2^p+3^q?
As of now, there is no known general rule or pattern for determining the primality of 2^p+3^q. It is a highly complex mathematical problem that is still being actively researched by mathematicians. However, there have been some significant discoveries and conjectures about the behavior of this formula.
4. What are some potential applications of understanding the primality of 2^p+3^q?
Understanding the primality of 2^p+3^q has potential applications in cryptography, as it is closely related to the security of certain encryption algorithms. It also has implications for understanding the distribution of prime numbers and their relationship with other mathematical concepts, which can lead to further advancements in number theory and other fields of mathematics.
5. Can the primality of 2^p+3^q be proven?
As of now, the primality of 2^p+3^q cannot be proven for all values of p and q. However, there have been some significant discoveries and conjectures about the behavior of this formula that have been proven to hold true for certain values. The search for a proof of its primality for all values continues to be an active area of research in mathematics.
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