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Difference between revisions of "Foot Prints Of God"

(→‎Infinitude of Foot Prints)
(→‎Foot Prints Of Primes)
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==Foot Prints Of Primes==
 
  
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The pattern in which the primes in the natural number line is an interesting topic and mathematicians are researching on the patterns of prime or the so called '''Footprints of Prints'''. Some mathematicians even designated it as the '''Foot Prints of God'''. There is no exact pattern found till date but have some nice facts and inequalities on them.
+
 
 +
The pattern in which the primes in the natural number line is an interesting topic and mathematicians are researching on the patterns of prime or the so called '''Footprints of Prints'''. Some mathematicians even designated it as the '''Foot Prints of God'''. It is also tempting to look for
 +
patterns amongst the primes: Can we find a formula that describes all of the primes? Or
 +
at least some of them? Are there actually infinitely many? And, if so, can we quickly
 +
determine how many there are up to a given point? Or at least give a good estimate?
 +
Once one has spent long enough determining primes, one cannot help but ask whether it
 +
is possible to recognize prime numbers quickly and easily? There is no exact pattern found till date but have some nice facts and inequalities on them.
  
 
==Infinitude of Foot Prints==
 
==Infinitude of Foot Prints==

Revision as of 03:31, 22 August 2012


The pattern in which the primes in the natural number line is an interesting topic and mathematicians are researching on the patterns of prime or the so called Footprints of Prints. Some mathematicians even designated it as the Foot Prints of God. It is also tempting to look for patterns amongst the primes: Can we find a formula that describes all of the primes? Or at least some of them? Are there actually infinitely many? And, if so, can we quickly determine how many there are up to a given point? Or at least give a good estimate? Once one has spent long enough determining primes, one cannot help but ask whether it is possible to recognize prime numbers quickly and easily? There is no exact pattern found till date but have some nice facts and inequalities on them.

Infinitude of Foot Prints

Euclidean Method

  • The first step towards these primes was probably taken by Euclid. He proved that these Foot Prints or primes are infinite in number. His method of proof was by contradiction. He firstly assumed that there are finitely many primes, say, ${p_1, p_2, p_3, ........ , p_n}$ and out of which $p_n$ is greatest. But now the number
                                            $N = \prod_{d=1}^{n}p_d + 1$

is not divisible by any of the assumed primes, it must be a prime itself. Also, $N > p_n$. Now N does not belong to the assumed set of primes but our assumption tells us that ${p_1, p_2, p_3, ........ , p_n}$ are the only primes. So contradiction$!$. Hence our assumption was wrong. Thus there is Infinitude of Primes or Foot Prints. Note:We could even prove in this way by taking $p$ as the highest prime of the assumed set and then taking $N = p! + 1$.

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