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Let Xn be the largest prime number.

and prime numbers less than Xn are X(n-1), X(n-2), X(n-3) .... X1

So, we can say that [ (Xn * X(n-1) * X(n-2) * ... * .......*1) + 1] is a number which isn't divisible by any of the prime numbers less than Xn.

Which means this is a new prime number.

Similarly, we can always have a prime number greater than a given prime numbers.

Thus, there is an infinite number of Prime Numbers.

Is the above proof OK?

Now please tell me

1.How do you prove that there are infinite Perfect numbers?

*A perfect number is a number which is half of the sum of its factors.

2.How do you prove that there are infinite Kaprekar numbers?

*A Kaprekar number is a number which is either of the following:

a)It is 1

b)The decimal representation of its square may be split once into two parts consisting of positive integers which sum to the original number. Note: that a split resulting in a part consisting purely of 0s is not valid, as 0 is not considered positive.

P.S.: I am excited about Kaprekar numbers because I contributed this part of code in rosettacode:

http://rosettacode.org/wiki/Kaprekar_numbers#Python

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 15,954

Hi Agnishom

The proof you gave doesn't go like that. You just messed up the last part. When you show that the number you got is a prime larger than Xn which is a contradiction, thus there is no largest prime i.e. there is an infinite number of primes.

1) It is still an open problem whether or not there is an infinite number of perfect numbers, and it is connected to the problem of infinite Mersenne primes.

2) I don't know if the number of Kaprekar numbers is infinite, but I think it is.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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One question.......

Four years ago, I remember writing a program to search for Perfect numbers.

I got 6, then 28, then a 3-digit-number, then 8128.

Then the program went on running but did not yield any more perfect numbers. Is it because I did not allow it to run for more than 15 minutes?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 15,954

The next perfect number is 33550336, so I guess that might be the reason you didn't get the next one. May we take a look at your code?

And the three digit number you are mentioning is 496.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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The code?

It was almost 5 years ago when I was in Standard 5 and I was only 10 years

I have changed my computer 2 times and formatted my Hard drive once within this gap of 5 years

Didn't care to have a backup

LOL

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'But our love is like the wind. I can't see it but I can feel it.' -A Walk to remember

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