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Proof:[n^n]-[(n-1)^(n-1)] is a prime n=natural number

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• Apple Master

Sorry to disappoint you but your assertion is false:

Posted on Apr 24, 2010

• k24674 Apr 27, 2010

It is cear from the screen capture that for n=9 9^9 -8^8 is divisible by 7, by 11 and by 13 because it is divisible by their product. So, you cannot declare that n^n-(n-1)^(n-1) is a prime number for all natural numbers.

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Posted on Jan 02, 2017

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Why is it important that 1 is not a prime number

It used to be. The idea is that a prime decomposition must be unique. If you write 13=1*13, you can also write 13=1*1*1*....*1*13. If you allow 1 to be a prime number, there is no unicity of any prime factor decomposition. Any prime number would have a infinity number of prime decompositions.

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I cant understand prime numbers

A prime number is a positive whole number (natural number) which cannot be divided into smaller whole numbers. In short, it can only be divided (integer division) by itself and 1.
First prime number is 2, then 3, 5, 7, 11, 13, 17, 19,
To generate a list of prime numbers the Greek Eratoshenes used a method called now the sieve Eratosthenes.

1. Write all the natural numbers up to some chosen limit (100, 259, any limit )
2. Remove 1 by crossing it ( 1 is no longer considered prime).
3. The first prime number is 2. Circle the number 2.
4. Go through the whole list crossing out all the numbers that are multiple of 2, that is 4, 6, 8,10,12, ...
5. Repeat.
6. The next prime number is 3 since it was not crossed out as a multiple of 2. Circle 3.
7. Go through the whole list crossing all the numbers that are multiples of 3, and which have not already been crossed out as multiples of 2 .
8. Number 4 has been crossed out already.
9. Next prime number is 5. Circle 5
10. Go through the whole list crossing all the numbers that are multiple of 5 and which have not been crossed out already as multiples of 2, or 3
11. Repeat:
12. Next prime number is 7. Circle 7
13. Go through the whole list again, crossing out the multiples of 7, and so on.
If you want to test the primality of a positive integer (odd number) starting by dividing it by 3 (use divisibility rules or otherwise), then divide by 5, by seven, and all subsequent prime numbers. If it is not divisible by any prime number, keep going but do not exceed the prime number that is closest yet smaller than square root of your number.

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What is a prime number

a prime number is a number that can only be multiplied by itself and 1 - example 3 can only be multiplied by itself and 1.

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One is not a prime why?

A prime numbers is defined as a natural number greater than one that has no positive divisors other than one and itself. Since one obviously is not greater than one, it is not a prime number.

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I dont know what all the prime numbers are

"A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself." - wikipedia
A list would be impossible without a limit. if you can't divide it by anything but 1 and the number itself(which would also mean that you were dividing by one), then it's prime.
4 is not prime, because it is divisible by 1, 2, and 4. 5 is prime, because it cannot be divided by anything except 1 and 5(itself).

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No, 1 is not a prime number. The decomposition of a number into prime factors must be unique so 2=1x2 is prime but since 1 can be written as 1=1x1x1x1....x1 with an arbitrary number of factors this opens up a can of worms as far as unicity of the decomposition is concerned. Conclusion: 1 has been ruled out as a prime number.
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