Greatest integer divides p^4 -1

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In summary, the greatest integer that divides p^4 - 1 for every prime number p greater than 5 is 240. This can be proven by factoring p^4 - 1 into (p^2 + 1)(p-1)(p+1) and showing that it is divisible by 16, 3, and 5. This makes it divisible by 16*3*5 = 240, and this holds true for all primes greater than 5.
  • #1
yxgao
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What is the greatest integer divides p^4 -1 for every prime number p greater than 5?

It is 240. Why?

Thanks!
 
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  • #2
Try factoring [itex]p^4-1[/itex].
 
  • #3
I already figured this out. (p^4-1) = (p^2+1)(p-1)(p+1).
i.) Each term is divisible by 2 since p is odd. Also, either p-1 or p+1 is divisible by 4.
So divisible by 16.
ii.) either p mod 3 = 1 or p mod 3 = 2. If the first case, p-1 = 0 mod 3, second case, p^2 -1 = 0 mod 3.
So divisible by 3.
iii.) either p mod 5 = 1, p mod 5 = 2, p mod 5 = 3, or p mod 5 = 4.
If p mod 5 = 1, p-1 = 0 mod 5. If p mod 4 = 1, p+1 = 0 mod 5, and if p mod 5 = 2 or p mod 3 = 1, then p^2+1 = 0 mod 5.
So divisible by 5.

Hence, divisible by 16*3*5 = 240.
 
  • #4
Er, my mistake, I misread the problem.

Well, you've verified that [itex]240 | p^4 - 1[/itex] for any prime number greater than 5, correct? (In fact, [itex]240 | n^4 - 1[/itex] if [itex](240, n) = 1[/itex])

The easiest way to proceed from here is, I think, to start looking at some explicit examples, and finish the proof from a small number of those. For instance, if [itex]m | p^4 - 1[/itex] for all primes [itex]p > 5[/itex], then [itex]m | 7^4 - 1 = 2400[/itex].
 

1. What is the significance of the greatest integer in the expression p^4 - 1?

The greatest integer, also known as the floor function, is used to round down a decimal number to the nearest integer. In this expression, it is used to ensure that the result is a whole number.

2. How does the value of p affect the result of p^4 - 1?

The value of p directly affects the result of p^4 - 1. As p increases, the result will also increase, and as p decreases, the result will also decrease. This is because raising a number to a higher power will result in a larger number.

3. Can any number be substituted for p in the expression p^4 - 1?

Yes, any real number can be substituted for p in this expression. However, if a non-integer value is used, the result may also be a decimal number, depending on the value of p.

4. What is the purpose of using the greatest integer in this expression?

The use of the greatest integer ensures that the result will always be a whole number. This is important in many mathematical calculations and can provide more accurate results in certain situations.

5. Are there any special cases for the expression p^4 - 1?

Yes, there are a few special cases for this expression. If p is equal to 0, the result will be -1. If p is equal to 1, the result will be 0. And if p is equal to -1, the result will be 2. These values can be confirmed by substituting them into the expression and using the greatest integer function.

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