MISCELLANEOUS RESULTS ON PRIME NUMBERS Many of the
... Theorem 0.6 (Wilson’s Theorem). Let p be an integer greater than 1. We have that p is prime if and only if (p − 1)! ≡ −1 (mod p). Proof. The result is clearly true if p = 2 or 3, so let us assume p > 3. If p is composite, then its positive divisors are among the integers 1, 2, 3, . . . , p − 1 and i ...
... Theorem 0.6 (Wilson’s Theorem). Let p be an integer greater than 1. We have that p is prime if and only if (p − 1)! ≡ −1 (mod p). Proof. The result is clearly true if p = 2 or 3, so let us assume p > 3. If p is composite, then its positive divisors are among the integers 1, 2, 3, . . . , p − 1 and i ...
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... Since f (z) has no zeros outside the polygon, we have, according to (2), to show only that the same concerns the second factor of the right hand side of (2). Let z be an arbitrary point outside the polygon. Because of its convexity, there is a line l through z such that the polygon is completely on ...
... Since f (z) has no zeros outside the polygon, we have, according to (2), to show only that the same concerns the second factor of the right hand side of (2). Let z be an arbitrary point outside the polygon. Because of its convexity, there is a line l through z such that the polygon is completely on ...
Lecture 5
... Constructible numbers. We begin with a rough definition which will be explained in more detail below. Definition (Constructible number - rough version). A real number α P R is constructible if we can construct a line segment of length |α| in a finite number of steps using from a fixed line segment o ...
... Constructible numbers. We begin with a rough definition which will be explained in more detail below. Definition (Constructible number - rough version). A real number α P R is constructible if we can construct a line segment of length |α| in a finite number of steps using from a fixed line segment o ...
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Popular values of Euler`s function
... §1. Introduction. For each natural number m, let N(m) denote the number of integers n with (f>(n) = m, where <> / denotes Euler's function. There are many interesting problems connected with the function N(m), such as the conjecture of Carmichael that N(m) is never 1 (see [9], for example) and the s ...
... §1. Introduction. For each natural number m, let N(m) denote the number of integers n with (f>(n) = m, where <> / denotes Euler's function. There are many interesting problems connected with the function N(m), such as the conjecture of Carmichael that N(m) is never 1 (see [9], for example) and the s ...
This phenomenon of primitive threes of Pythagoras owes it`s
... PACS numbers: 02.10.Ab , 02.30.Xx In book " The last theorem of P.Fermat" by G. Edwards» ,translated from English into Russia and published by "Mir" publishing House in 1980 in Moscow, we read on page 14 (see also [2] ) : «Written in Latin language article by Fermat says that “from the other hand it ...
... PACS numbers: 02.10.Ab , 02.30.Xx In book " The last theorem of P.Fermat" by G. Edwards» ,translated from English into Russia and published by "Mir" publishing House in 1980 in Moscow, we read on page 14 (see also [2] ) : «Written in Latin language article by Fermat says that “from the other hand it ...
![[Part 1]](http://s1.studyres.com/store/data/008795996_1-7bdba077dfd2123ff356afe25da5d3ed-300x300.png)
[Part 1]
... TRANSCEHDESITAL HUMBEKS BASED OH THE FIBONACCI SEQUENCE DONALD KNUTH California Institute of Technology, Pasadena, California ...
... TRANSCEHDESITAL HUMBEKS BASED OH THE FIBONACCI SEQUENCE DONALD KNUTH California Institute of Technology, Pasadena, California ...
Math 248, Methods of Proof, Winter 2015
... 3. Prove (by contradiction) that there does not exists a smallest positive real number (that is there does not exists an r ∈ R such that r > 0 and, if s ∈ R and s > 0 then r ≤ s). Sometimes we will want to prove that a statement of the form (∀x)(P (x)) is false. If we do this by giving a constructiv ...
... 3. Prove (by contradiction) that there does not exists a smallest positive real number (that is there does not exists an r ∈ R such that r > 0 and, if s ∈ R and s > 0 then r ≤ s). Sometimes we will want to prove that a statement of the form (∀x)(P (x)) is false. If we do this by giving a constructiv ...
Full text
... Then a0 = 1, and v(an) = S(n) for n>0. Our main result, Theorem 4.1, generalizes Theorem 1.1 to series of the form [H(2kx)J, where r is rational and 2k (k > 1) is the highest power of 2 dividing the denominator of r. A summary by sections follows. Section 2 is a preliminary section in which we state ...
... Then a0 = 1, and v(an) = S(n) for n>0. Our main result, Theorem 4.1, generalizes Theorem 1.1 to series of the form [H(2kx)J, where r is rational and 2k (k > 1) is the highest power of 2 dividing the denominator of r. A summary by sections follows. Section 2 is a preliminary section in which we state ...
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... a. Write a recurrence formula which gives the amount of time T (n ) wasted by this algorithm. b. Use the Master Theorem to find an asymptotic solution to this recurrence. 9. Prove that all trees on n vertices have n 1 edges. Do this by (a) induction on the number of vertices, and (b) by induction ...
... a. Write a recurrence formula which gives the amount of time T (n ) wasted by this algorithm. b. Use the Master Theorem to find an asymptotic solution to this recurrence. 9. Prove that all trees on n vertices have n 1 edges. Do this by (a) induction on the number of vertices, and (b) by induction ...
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... Theorem (Nicomachus). The sum of the cubes of the first n integers is equal to the square of the nth triangular number. To put it algebraically, n X ...
... Theorem (Nicomachus). The sum of the cubes of the first n integers is equal to the square of the nth triangular number. To put it algebraically, n X ...
Full text
... 1. INTRODUCTION A well-known digital expansion is the so-called Zeckendorf number system [7], where every positive integer n can be written as L k=Q ...
... 1. INTRODUCTION A well-known digital expansion is the so-called Zeckendorf number system [7], where every positive integer n can be written as L k=Q ...
Class notes from November 18
... From the above example we see that one can find the decryption exponent fairly quickly once we’ve found the prime factorization of N. Even for very large values of N (several hundred digits long), a computer can efficiently implement the Euclidean algorithm to find the inverse of an element once the ...
... From the above example we see that one can find the decryption exponent fairly quickly once we’ve found the prime factorization of N. Even for very large values of N (several hundred digits long), a computer can efficiently implement the Euclidean algorithm to find the inverse of an element once the ...
A NOTE ON AN ADDITIVE PROPERTY OF PRIMES 1. Introduction
... and the size of elements of the set A and the fact that all the possible subsets must have different sums. This theorem can be proved,in various ways. One of them, as one can find in [1], uses High School Algebra only, and so it can be regarded as fully elementary. 4. The elementary proof of Theorem ...
... and the size of elements of the set A and the fact that all the possible subsets must have different sums. This theorem can be proved,in various ways. One of them, as one can find in [1], uses High School Algebra only, and so it can be regarded as fully elementary. 4. The elementary proof of Theorem ...
Math311W08Day3
... we can dismiss these with scorn and derision, beating them into submission with our sledgehammer of a theorem! Lemma 2.11: If an → a then for any number α, αan → αa. Lemma 2.12: If an and bn both converge to zero, then so does an bn. For practice, try to prove these directly (they are in the book so ...
... we can dismiss these with scorn and derision, beating them into submission with our sledgehammer of a theorem! Lemma 2.11: If an → a then for any number α, αan → αa. Lemma 2.12: If an and bn both converge to zero, then so does an bn. For practice, try to prove these directly (they are in the book so ...
Euler`s totient function and Euler`s theorem
... Definition 2.2. Let n ≥ 1. A set of ϕ(n) integers such that (i) each is relatively prime to n, and (ii) any two distinct members are incongruent modulo n is called a reduced system of residues modulo n. Example 2.3. The canonical way to think of a reduced system of residues modulo an integer n ≥ 1 i ...
... Definition 2.2. Let n ≥ 1. A set of ϕ(n) integers such that (i) each is relatively prime to n, and (ii) any two distinct members are incongruent modulo n is called a reduced system of residues modulo n. Example 2.3. The canonical way to think of a reduced system of residues modulo an integer n ≥ 1 i ...