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wsldam
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Prove that 3n + 1 has an odd prime divisor for all natural numbers > 1. I tried using order but it didn't really get me anywhere. Would prefer hints rather than complete solutions. Thanks.
3^n+1 has an odd prime divisor
- Thread starter wsldam
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In summary, to prove that 3n + 1 has an odd prime divisor for all natural numbers > 1, start by playing around with the numbers and noticing patterns. One approach is to work out the values of a few numbers, 32+1, 33+1, 34+1, ... Another approach is to prove that 2m can never be of the form 3n + 1. This can be done by either considering it modulo 8 or by considering cases of m being odd or even.
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AlephZero
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If you can't see how to prove this sort of result, start by playing around with the numbers.
Work out the values of a few numbers, 32+1, 33+1, 34+1, ... What do you notice about them?
Work out the values of a few numbers, 32+1, 33+1, 34+1, ... What do you notice about them?
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tiny-tim
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welcome to pf!
hi wsldam! welcome to pf!
isn't that another way of saying that 2m can never be of the form 3n + 1 ?
hi wsldam! welcome to pf!
wsldam said:
isn't that another way of saying that 2m can never be of the form 3n + 1 ?
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AlephZero
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tiny-tim said:hi wsldam! welcome to pf!
isn't that another way of saying that 2m can never be of the form 3n + 1 ?
It is also saying that 3n + 1 is never an odd prime (but that's easy to show).
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Norwegian
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solution 1: Do it modulo 8
solution 2: Do it for m odd (mod 3) and m even (factor 2m-1).
solution 2: Do it for m odd (mod 3) and m even (factor 2m-1).
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tiny-tim
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nice!
1. What is the significance of "3^n+1" in this statement?
The expression "3^n+1" refers to a mathematical sequence in which n is a positive integer. This sequence is often studied in number theory and has connections to prime numbers.
2. Why is it important for "3^n+1" to have an odd prime divisor?
Having an odd prime divisor in the expression "3^n+1" is significant because it helps to reveal patterns and properties of the sequence. It also has implications for understanding the distribution of prime numbers.
3. Can you provide an example of "3^n+1" with an odd prime divisor?
Yes, for n=2, the expression "3^n+1" becomes "3^2+1=10". This has an odd prime divisor of 5.
4. Is there a specific formula for determining when "3^n+1" has an odd prime divisor?
There is no general formula for determining when "3^n+1" has an odd prime divisor. However, there are certain patterns and criteria that have been identified by mathematicians, such as the Lucas-Lehmer test, which can be used to determine if a specific value of n will result in an odd prime divisor.
5. What are the potential applications of studying "3^n+1" and its odd prime divisors?
Studying "3^n+1" and its odd prime divisors can have various implications in number theory and other branches of mathematics. It has been used in the study of Mersenne primes, which are prime numbers of the form "2^n-1". It also has connections to the Collatz conjecture, a famous unsolved problem in mathematics. Additionally, understanding the distribution of prime numbers can have practical applications in cryptography and computer science.
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