How to Submit Solutions for POTW #330?

  • MHB
  • Thread starter Euge
  • Start date
In summary, the guidelines for submitting a solution to POTW #330 are as follows: The deadline for submitting a solution is typically one week after the problem is posted on the website. Solutions submitted after the deadline will not be considered for evaluation. There is no specific word/character limit for the solution, but it is recommended to keep it concise and to the point. There are no specific formatting requirements, but it is recommended to use a clear and organized structure for the solution. The solutions will be evaluated and judged based on accuracy, clarity, and creativity, as well as the use of proper scientific methods and reasoning.
  • #1
Euge
Gold Member
MHB
POTW Director
2,073
244
Here is this week's POTW:

-----
For $n$ a positive integer, let $\phi(n)$ be the number of positive integers not exceeding $n$ and coprime to $n$. Show that every composite number $n \equiv 1\pmod{\phi(n)}$ has at least three distinct prime divisors. (In fact, $n$ would have at least four distinct prime divisors. You can prove this harder result if you like.)
-----

Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!
 
Physics news on Phys.org
  • #2
No one answered this week's problem. You can read my solution below.
By way of contradiction, suppose $n$ has only two distinct prime factors. Since $n \equiv 1 \pmod{\phi(n)}$, $n$ is not divisible by the square of any prime, and thus $n = p_1 p_2$ for some distinct primes $p_1, p_2$. Consider the integer
$$\frac{n-1}{\phi(n)} = \frac{p_1p_2 -1}{(p_1 - 1)(p_2 - 1)} = 1 + \frac{1}{p_1 - 1} + \frac{1}{p_2 - 1}$$ The latter expression is greater than $1$ and no greater than $2.5$, so being an integer
$$1 + \frac{1}{p_1 - 1} + \frac{1}{p_2 - 1} = 2$$
This forces $p_1 = p_2 = 3$, contradicting the assumption that $p_1 \neq p_2$.
 

FAQ: How to Submit Solutions for POTW #330?

What are the Guidelines for Submitting a Solution to POTW #330?

The guidelines for submitting a solution to POTW #330 are as follows:

  • 1. What is the deadline for submitting a solution?
  • 2. Can I submit a solution after the deadline?
  • 3. Is there a word/character limit for the solution?
  • 4. Are there any specific formatting requirements?
  • 5. How will the solutions be evaluated and judged?

1. The deadline for submitting a solution to POTW #330 is typically one week after the problem is posted on the website.

2. No, solutions submitted after the deadline will not be considered for evaluation.

3. There is no specific word/character limit for the solution, but it is recommended to keep it concise and to the point.

4. There are no specific formatting requirements, but it is recommended to use a clear and organized structure for the solution.

5. The solutions will be evaluated and judged based on accuracy, clarity, and creativity. The judges will also take into consideration the use of proper scientific methods and reasoning.

Back
Top