Difference between revisions of "Foot Prints Of God"
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*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, <math>\{p_1, p_2, p_3, ........ , p_n}</math> and out of which <math>p_n</math> is greatest. But now the number <math>N = \prod_{d=1}^{n}p_d + 1</math> is not divisible by any of the assumed primes, it must be a prime itself. Also, <math>N > p_n</math>. Now N does not belong to the assumed set of primes but our assumption tells us that <math>\{p_1, p_2, p_3, ........ , p_n}</math> are the only primes. So contradiction<math>!</math>. Hence our assumption was wrong. Thus there is Infinitude of Primes or Foot Prints. | *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, <math>\{p_1, p_2, p_3, ........ , p_n}</math> and out of which <math>p_n</math> is greatest. But now the number <math>N = \prod_{d=1}^{n}p_d + 1</math> is not divisible by any of the assumed primes, it must be a prime itself. Also, <math>N > p_n</math>. Now N does not belong to the assumed set of primes but our assumption tells us that <math>\{p_1, p_2, p_3, ........ , p_n}</math> are the only primes. So contradiction<math>!</math>. Hence our assumption was wrong. Thus there is Infinitude of Primes or Foot Prints. | ||
| − | We could even prove in this way by taking <math>p</math> as the highest prime of the assumed set and then taking <math>N = p! + 1</math>. | + | We could even prove in this way by taking <math>p</math> as the highest prime of the assumed set and then taking |
| + | <math>N = p! + 1</math>. | ||
Revision as of 13:03, 18 August 2012
Foot Prints Of Primes
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.
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, $\{p_1, p_2, p_3, ........ , p_n}$ (Error compiling LaTeX. Unknown error_msg) and out of which
is greatest. But now the number 
is not divisible by any of the assumed primes, it must be a prime itself. Also, 
. Now N does not belong to the assumed set of primes but our assumption tells us that $\{p_1, p_2, p_3, ........ , p_n}$ (Error compiling LaTeX. Unknown error_msg) are the only primes. So contradiction
. Hence our assumption was wrong. Thus there is Infinitude of Primes or Foot Prints.
is greatest. But now the number 
is not divisible by any of the assumed primes, it must be a prime itself. Also, 
. Now N does not belong to the assumed set of primes but our assumption tells us that 
. Hence our assumption was wrong. Thus there is Infinitude of Primes or Foot Prints.We could even prove in this way by takingas the highest prime of the assumed set and then taking
as the highest prime of the assumed set and then taking
.
