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Number Theory Prime Number Theorem

matqkks

Member
Jun 26, 2012
74
I am looking for resources which explain the prime number theorem to 18 year old students. I am not seeking a proof of the result but something which will have an impact and motivate a student to study mathematics in the future. Can anyone provide or direct me to these resources?
 

chisigma

Well-known member
Feb 13, 2012
1,704
I am looking for resources which explain the prime number theorem to 18 year old students. I am not seeking a proof of the result but something which will have an impact and motivate a student to study mathematics in the future. Can anyone provide or direct me to these resources?
The prime number theorem is related to the Prime Counting Function that is defined as 'the number of prime numbers less than or equal to some real number x' and is denoted as $\displaystyle \pi(x)$. When hi was fourteen the German mathematician Karl Friedrik Gauss analysed the problem and, observing the probabilistic distribution of the primes, given a number n, the probability that n was prime was approximatively equal to $\displaystyle p(n) \sim \frac{1}{\ln n}$ and from that it derives that is...

$\displaystyle \pi(x) \sim \frac{x}{\ln x}\ (1)$

In the last two centuries great work has be done about this problem and one of the most remarkable result is the Prime Number Theorem that, in a certain sense, confirms the 'discovery' of Gauss extablishing that is...

$\displaystyle \lim_{ x \rightarrow \infty} \frac{\pi(x)}{\frac{x}{\ln x}} = 1\ (2)$

Kind regards

$\chi$ $\sigma$
 

mathbalarka

Well-known member
MHB Math Helper
Mar 22, 2013
573
The original theorem of Chebyshev was -- there exists some A, B > 0 such that for all x > 2,

\(\displaystyle \frac{Ax}{\log x} < \pi(x) < \frac{Bx}{\log x}\)

It was later established by Hadamard and de la Vallée Poussin the one shown by chisigma in the previous post.

PS I think there is nothing in the proof of PNT that cannot be understandable to an 18 year onld.
 

chisigma

Well-known member
Feb 13, 2012
1,704
... I think there is nothing in the proof of PNT that cannot be understandable to an 18 year old...
It depends from how many 'neurons' the young boy has... for example the fellow in the figure below, when hi was thirteen, was capable to reproduce the Allegri's Miserere, rigorously taken as a 'copyright' from Vatican, after having heard it onle one time...


Wolfgang-Mozart-9417115-2-402.jpg

... in December, 1769, Wolfgang, then age 13, and his father departed from Salzburg for Italy, leaving his mother and sister at home. It seems that by this time Nannerl’s professional music career was over. She was nearing marriageable age and according to the custom of the time, she was no longer permitted to show her artistic talent in public. The Italian outing was longer than the others (1769-1771) as Leopold wanted to display his son’s abilities as a performer and composer to as many new audiences as possible. While in Rome, Wolfgang heard Gregorio Allegri’s Miserere performed once in the Sistine Chapel. He wrote out the entire score from memory, returning only to correct a few minor errors. During this time Wolfgang also wrote a new opera, Mitridate, re di Ponto for the court of Milan. Other commissions followed and in subsequent trips to Italy, Wolfgang wrote two other operas, Ascanio in Alba (1771) and Lucio Silla (1772)...

Kind regards

$\chi$ $\sigma$
 

awkward

Member
Feb 18, 2012
36
In "Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics", John Derbyshire introduces the prime number theorem by displaying a table of N, ln(N), and N / pi(N), where pi(N) is the number of primes less than or equal to N, for some large values of N, ranging from 10^3 to 10^18. The numerical evidence that ln(N) is close to N / pi(N) is then strong. Maybe you could do something similar.

I highly recommend the book if you haven't read it, by the way.
 

chisigma

Well-known member
Feb 13, 2012
1,704
In "Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics", John Derbyshire introduces the prime number theorem by displaying a table of N, ln(N), and N / pi(N), where pi(N) is the number of primes less than or equal to N, for some large values of N, ranging from 10^3 to 10^18. The numerical evidence that ln(N) is close to N / pi(N) is then strong. Maybe you could do something similar.

I highly recommend the book if you haven't read it, by the way.
I wonder why a theorem that has been rigolusly demonstraded more that hundred years ago by Jaques Hadamard and Charles-Jean Étienne Gustave Nicolas de la Vallée Poussin needs today of 'numerical validations'... it is better for us to spend our time to perform more mathematical advances... or not?...

Kind regards

$\chi$ $\sigma$
 

mathbalarka

Well-known member
MHB Math Helper
Mar 22, 2013
573
akward said:
n "Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics", John Derbyshire introduces the prime number theorem by displaying a table of N, ln(N), and N / pi(N), where pi(N) is the number of primes less than or equal to N, for some large values of N, ranging from 10^3 to 10^18. The numerical evidence that ln(N) is close to N / pi(N) is then strong. Maybe you could do something similar.

I highly recommend the book if you haven't read it, by the way.
Actually \(\displaystyle x/log(x)\) is nothing more than a match of the order of magnitude. A far better (unconditional) approximation is the one that uses that zeros of zeta which is impressive if illustrated neatly (although I don't think any exists since evaluation of Li at complex values and manipulating that much zeros of zeta is very tiresome).

Balarka
.
 

eddybob123

Active member
Aug 18, 2013
76

awkward

Member
Feb 18, 2012
36
I wonder why a theorem that has been rigolusly demonstraded more that hundred years ago by Jaques Hadamard and Charles-Jean Étienne Gustave Nicolas de la Vallée Poussin needs today of 'numerical validations'... it is better for us to spend our time to perform more mathematical advances... or not?...

Kind regards

$\chi$ $\sigma$
From the OP:
"I am not seeking a proof of the result but something which will have an impact and motivate a student to study mathematics in the future."