| Stanisław Knapowski | |
| Birth Date: | 19 May 1931 |
| Birth Place: | Poznań, Poland |
| Death Place: | Florida, USA |
| Citizenship: | Polish |
| Fields: | Mathematician |
| Education: | Poznań University, Wrocław University and Adam Mickiewicz University |
| Known For: | Prime numbers and number theory |
| Awards: | Mazurkiewicz Prize, Rockefeller Scholarship |
Stanisław Knapowski (May 19, 1931 – September 28, 1967) was a Polish mathematician who worked on prime numbers and number theory. Knapowski published 53 papers despite dying at only 36 years old.
Stanisław Knapowski was the son of Zofia Krysiewicz and Roch Knapowski. His father, Roch Knapowski was a lawyer in Poznań but later taught at Poznań University. The family moved to the Kielce province in south-eastern Poland after the German invasion of 1939 but returned to Poznań after the war.[1]
Stanisław completed his high school education in 1949 excelling at math and continued on at Poznań University to study mathematics. Later in 1952 he continued his studies at University of Wrocław and earned his master's degree in 1954.
Knapowski was appointment an assistant at Adam Mickiewicz University in Poznań under Władysław Orlicz and worked towards his doctorate. He studied under the direction of Pál Turán starting in Lublin in 1956. He published many of his papers with Turán and Turán wrote a short biography of his life and work in 1971 after his death.[2] Knapowski began to work in this area and finished his doctorate in 1957 “Zastosowanie metod Turaná w analitycznej teorii liczb” ("Certain applications of Turan's methods in the analytical theory of numbers").
Knapowski spent a year in Cambridge where he worked with Louis J. Mordell and listened to classes by J.W.S. Cassels and Albert Ingham. He visited Belgium, France and The Netherlands.
Knapowski returned to Poznań to finish another thesis to complete a post-doctoral qualification needed to lecture at a German university. "On new "explicit formulas" in prime number theory" in 1960.[3] In 1962 the Polish Mathematical Society awarded him their Mazurkiewicz Prize and he moved to Tulane University in New Orleans, United States. After a very short return to Poland, he left again and taught in Marburg in Germany, Gainesville, Florida and Miami, Florida.[4]
Knapowski was a good classical pianist. He was an avid driver. He died in a traffic accident where he lost control of his car while leaving the Miami airport.
Knapowski expanded on the work of others in several fields of number theory, prime number theorem, modular arithmetic and non-Euclidean geometry.
See main article: Prime number theorem and Prime-counting function.
Mathematicians work on primality tests to develop easier ways to find prime numbers when finding them by trial division is not practical. This has many applications in cybersecurity. There is no formula to calculate prime numbers. However, the distribution of primes can be statistically modelled. The prime number theorem, which was proven at the end of the 19th century, says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits (logarithm). At the start of the 19th century, Adrien-Marie Legendre and Carl Friedrich Gauss suggested that as
x
x
x/logx
logx
x
\operatorname{li}(n)=
| n | |
| \int | |
| 0 |
| dt | |
| logt |
where the integral is evaluated at
t=n
\pi(n)
n
And
\Delta(n)=\pi(n)-\operatorname{li}(n)
Bernhard Riemann stated that
\Delta(n)
\pi(x)>\operatorname{Li}(x)+
| ||||
| logloglog |
x,
and that there are also arbitrarily large values of x for which
\pi(x)<\operatorname{Li}(x)-
| ||||
| logloglog |
x.
Thus the difference (x) − Li(x) changes sign infinitely many times.
x
\pi(x)>\operatorname{li}(x),
Knapowski followed this up and published a paper on the number of times
\Delta(n)
\Delta(n)
See main article: Modular arithmetic.
k
Modular arithmetic modifies usual arithmetic by only using the numbers
\{0,1,2,...,n-1\}
n
n
The distribution of the primes looks random, without a pattern. Take a list of consecutive prime numbers and divide them by another prime (like 7) and keep only the remainder (this is called reducing them modulo 7). The result is a sequence of integers from 1 to 6. Knapowski worked to determine the parameters of this modular distribution