Difference between revisions of "Foot Prints Of God"
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| − | + | The so-far undiscovered pattern in which the [[prime number|primes]] in the [[number line]] lie is called the '''Footprints of Primes'''. 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, <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> | |
| − | 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> | + | 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'''. |
| − | + | '''Note:'''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>. | |
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Latest revision as of 17:15, 14 April 2017
The so-far undiscovered pattern in which the primes in the number line lie is called the Footprints of Primes. 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,
and out of which 
is greatest. But now the number
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 
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 
as the highest prime of the assumed set and then taking 
.
