The sums of the reciprocals of Fibonacci polynomials and Lucas
... © 2012 Zhengang and Wenpeng; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor ...
... © 2012 Zhengang and Wenpeng; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor ...
Diskrete Mathematik für Informatik (SS 2017)
... of discrete mathematics which are of importance for computer science. I would like to warn you explicitly not to regard these slides as the sole source of information on the topics of my course. It may and will happen that I’ll use the lecture for talking about subtle details that need not be covere ...
... of discrete mathematics which are of importance for computer science. I would like to warn you explicitly not to regard these slides as the sole source of information on the topics of my course. It may and will happen that I’ll use the lecture for talking about subtle details that need not be covere ...
1. Modular arithmetic
... Numbers with this property (ap = a mod p for all 1 a p 1) are called Carmichael numbers. Eg. 561, 1105, 1729, 2465, 2821, 6601, 8911 are all the Carmichael numbers less than 10; 000. So maybe these numbers are very rare and this is not a problem. Exercise 6.2. Can you show that there are in nite ...
... Numbers with this property (ap = a mod p for all 1 a p 1) are called Carmichael numbers. Eg. 561, 1105, 1729, 2465, 2821, 6601, 8911 are all the Carmichael numbers less than 10; 000. So maybe these numbers are very rare and this is not a problem. Exercise 6.2. Can you show that there are in nite ...
An introduction to the Smarandache Square
... Case 1. According to the theorem 7 Ssc(n)=n and Ssc(n+1)=n+1 that implies that Ssc(n)<>Ssc(n+1) Case 2. Without loss of generality let's suppose that: n = pa ⋅ q b n + 1 = p a ⋅ qb + 1 = sc ⋅ t d where p,q,s and t are distinct primes. According to the theorem 4: Ssc( n) = Ssc ( p a ⋅ q b ) = p odd ( ...
... Case 1. According to the theorem 7 Ssc(n)=n and Ssc(n+1)=n+1 that implies that Ssc(n)<>Ssc(n+1) Case 2. Without loss of generality let's suppose that: n = pa ⋅ q b n + 1 = p a ⋅ qb + 1 = sc ⋅ t d where p,q,s and t are distinct primes. According to the theorem 4: Ssc( n) = Ssc ( p a ⋅ q b ) = p odd ( ...
the infinity of the twin primes
... 5 x 3 x 5). It can be the product of a twin primes pair with itself and another prime or primes (e.g., 3 x 5 x 3 x 5 x 23 x 89). It can be the product of a twin primes pair with another twin primes pair or other twin primes pairs (e.g., 11 x 13 x 227 x 229 x 461 x 463) It can be the product of a tw ...
... 5 x 3 x 5). It can be the product of a twin primes pair with itself and another prime or primes (e.g., 3 x 5 x 3 x 5 x 23 x 89). It can be the product of a twin primes pair with another twin primes pair or other twin primes pairs (e.g., 11 x 13 x 227 x 229 x 461 x 463) It can be the product of a tw ...
![arXiv:math/0604314v2 [math.NT] 7 Sep 2006 On](http://s1.studyres.com/store/data/016163041_1-f0b243339013698f62e5d1397917ed3b-300x300.png)
arXiv:math/0604314v2 [math.NT] 7 Sep 2006 On
... Theorem 4 The only squarefull integers not in R are 1, 4, 8, 9, 16 and 36. We recall that an integer n is said to be squarefull if for every prime divisor p of n we have p2 |n. An integer n is called t-free if pt ∤ m for every prime number p. (Thus saying a number is squarefree is the same as saying ...
... Theorem 4 The only squarefull integers not in R are 1, 4, 8, 9, 16 and 36. We recall that an integer n is said to be squarefull if for every prime divisor p of n we have p2 |n. An integer n is called t-free if pt ∤ m for every prime number p. (Thus saying a number is squarefree is the same as saying ...
Module - Exponents
... 2. Use laws of exponents to simplify expressions with zero or negative exponents. 3. Use laws of exponents to simplify expressions with rational exponents. ...
... 2. Use laws of exponents to simplify expressions with zero or negative exponents. 3. Use laws of exponents to simplify expressions with rational exponents. ...
Cryptography and Number Theory
... For thousands of years people have searched for ways to send messages secretly. There is a story that, in ancient times, a king needed to send a secret message to his general in battle. The king took a servant, shaved his head, and wrote the message on his head. He waited for the servant’s hair to g ...
... For thousands of years people have searched for ways to send messages secretly. There is a story that, in ancient times, a king needed to send a secret message to his general in battle. The king took a servant, shaved his head, and wrote the message on his head. He waited for the servant’s hair to g ...
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The theretofore unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".