Find Fractional Part of $(p+1)!/(p^2)$

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In summary, "Find Fractional Part of $(p+1)!/(p^2)$" means finding the decimal part of the value obtained by dividing the factorial of (p+1) by the square of p. A factorial is a mathematical operation denoted by an exclamation mark (!) and the fractional part of a number is the decimal portion after the decimal point. To find the fractional part of a number, simply subtract the integer part of the number from the original number. To find the fractional part of $(p+1)!/(p^2)$, first calculate (p+1)! and then divide it by p^2 and the remainder obtained from this division will be the fractional part of the expression.
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anemone
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Let $p$ be a prime number. Find the fractional part of $\dfrac{(p+1)!}{p^2}$.
 
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Because p is prime by wilson theorem $p | (p-1)!+1$
Or $p^2| (p! + p)$
or $p^2| (p! + p) (p+1)$
Or $p^2| (p+1)! + p(p+1)$
or $(p+1)! \equiv -p(p+1) \pmod p^2$

So fractional part of $\frac{(p+1)!}{p^2}$ is same as fractional part of $\frac{-p(p+1)}{p^2}$ or is $\frac{p-1}{p}$
 

1. What is the purpose of finding the fractional part of $(p+1)!/(p^2)$?

The purpose of finding the fractional part of $(p+1)!/(p^2)$ is to determine the decimal representation of the resulting fraction. This can be useful in various mathematical calculations and applications.

2. How is the fractional part of $(p+1)!/(p^2)$ calculated?

The fractional part of $(p+1)!/(p^2)$ is calculated by dividing the numerator by the denominator and taking the remainder. This remainder represents the fractional part of the resulting fraction.

3. Can the fractional part of $(p+1)!/(p^2)$ be simplified?

Yes, the fractional part of $(p+1)!/(p^2)$ can be simplified by dividing both the numerator and denominator by their greatest common factor.

4. What is the range of values for the fractional part of $(p+1)!/(p^2)$?

The range of values for the fractional part of $(p+1)!/(p^2)$ is between 0 and 1, inclusive. This means that the resulting fraction will always have a decimal value between 0 and 1.

5. Are there any special cases to consider when finding the fractional part of $(p+1)!/(p^2)$?

Yes, there are a few special cases to consider when finding the fractional part of $(p+1)!/(p^2)$. These include when the numerator is a multiple of the denominator, resulting in a fractional part of 0, and when the denominator is 0, resulting in an undefined fractional part.

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