MATH 351 – FOM HOMEWORK 1. Solutions A. Statement: √ 2 is
... These statements are not logically equivalent. F says that S has a maximum element, namely z. G says that every element has another element bigger than it. So, here is an example of a set that satisfies G and not F : S = {x ∈ Q|x < π}. Notice that since we can approximate π with rational numbers, ...
... These statements are not logically equivalent. F says that S has a maximum element, namely z. G says that every element has another element bigger than it. So, here is an example of a set that satisfies G and not F : S = {x ∈ Q|x < π}. Notice that since we can approximate π with rational numbers, ...
Sharp estimate on the supremum of a class of sums of small i.i.d.
... We show that if N ≥ n201 and n ≥ 41L, then the above model satisfies the conditions of Theorem 1, and compare the bound we got for Pu,n in our previous calculation with the estimate of Theorem 1 in this example. We shall apply Theorem 1 for the class of functions F consisting of the indicator functi ...
... We show that if N ≥ n201 and n ≥ 41L, then the above model satisfies the conditions of Theorem 1, and compare the bound we got for Pu,n in our previous calculation with the estimate of Theorem 1 in this example. We shall apply Theorem 1 for the class of functions F consisting of the indicator functi ...
Prime Numbers
... combination of the original numbers, it’s rather tedious. Here’s a better way. I’ll write it more formally, since the steps are a little complicated. Terminology. If a and b are things, a linear combination of a and b is something of the form sa + tb, where s and t are numbers. (The kind of “number” ...
... combination of the original numbers, it’s rather tedious. Here’s a better way. I’ll write it more formally, since the steps are a little complicated. Terminology. If a and b are things, a linear combination of a and b is something of the form sa + tb, where s and t are numbers. (The kind of “number” ...
36(4)
... Since Fn mdFn+l are coprime integers, and because any prime/? divides infinitely many Fibonacci numbers Fn_l = Fn+l-Fn, for every prime p the congruence Fn = Fn+l # 0 modp is satisfied for infinitely many pairs of Fibonacci numbers Fn and Fn+l. Proof of Theorem 1.2: Let 36 c: (0,1) be the subset of ...
... Since Fn mdFn+l are coprime integers, and because any prime/? divides infinitely many Fibonacci numbers Fn_l = Fn+l-Fn, for every prime p the congruence Fn = Fn+l # 0 modp is satisfied for infinitely many pairs of Fibonacci numbers Fn and Fn+l. Proof of Theorem 1.2: Let 36 c: (0,1) be the subset of ...
Variant of a theorem of Erdős on the sum-of-proper
... CARL POMERANCE AND HEE-SUNG YANG Abstract. In 1973, Erdős proved that a positive proportion of numbers are not of the form σ(n) − n, the sum of the proper divisors of n. We prove the analogous result where σ is replaced with the sum-of-unitary-divisors function σ ∗ (which sums divisors d of n such ...
... CARL POMERANCE AND HEE-SUNG YANG Abstract. In 1973, Erdős proved that a positive proportion of numbers are not of the form σ(n) − n, the sum of the proper divisors of n. We prove the analogous result where σ is replaced with the sum-of-unitary-divisors function σ ∗ (which sums divisors d of n such ...
37(2)
... Also, it is clear that G is bijective and x < G(x) for all x > 0. Thus, G~l exists and is also strictly increasing with G~l(x) < x. Let u = G_1(x). Then G(u) = x and u = 3x + v8x 2 +1. Since u < x, we have u = 3x - V8x2 4-1. Also, since 8(G_1(x))2 +1 = (8x - 3V8x2 +1) 2 is a perfect square, it follo ...
... Also, it is clear that G is bijective and x < G(x) for all x > 0. Thus, G~l exists and is also strictly increasing with G~l(x) < x. Let u = G_1(x). Then G(u) = x and u = 3x + v8x 2 +1. Since u < x, we have u = 3x - V8x2 4-1. Also, since 8(G_1(x))2 +1 = (8x - 3V8x2 +1) 2 is a perfect square, it follo ...
A007970: Proof of a Theorem Related to the Happy Number
... . D and E are both odd if d is odd (in fact ≡ 3 (mod 4)), and they are U0 both even if d is even (in fact ≡ 0 (mod 4)). The solutions U0 , T0 are the minimal positive ones. Proof: One uses the basic result that the Pell eq. (2) has a fundamental positive solution (x0 , y0 ) for each non-square posit ...
... . D and E are both odd if d is odd (in fact ≡ 3 (mod 4)), and they are U0 both even if d is even (in fact ≡ 0 (mod 4)). The solutions U0 , T0 are the minimal positive ones. Proof: One uses the basic result that the Pell eq. (2) has a fundamental positive solution (x0 , y0 ) for each non-square posit ...
Here - UBC Math
... are then viewed modulo 26 and are enciphered via the enciphering transformation x 7→ ax + b (mod 26) . A transformation of the above form is a legitimate (i.e., one-to-one and onto) enciphering transformation if and only if gcd(a, 26) = 1. We say that a letter with numerical value x is “fixed” if x ...
... are then viewed modulo 26 and are enciphered via the enciphering transformation x 7→ ax + b (mod 26) . A transformation of the above form is a legitimate (i.e., one-to-one and onto) enciphering transformation if and only if gcd(a, 26) = 1. We say that a letter with numerical value x is “fixed” if x ...
(pdf)
... An integer is called a quadratic residue modulo p if it is congruent to a perfect square modulo p. The Legendre symbol, or quadratic character, tells us whether an integer is a quadratic residue or not modulo a prime p. The Legendre symbol has useful properties, such as multiplicativity, which can s ...
... An integer is called a quadratic residue modulo p if it is congruent to a perfect square modulo p. The Legendre symbol, or quadratic character, tells us whether an integer is a quadratic residue or not modulo a prime p. The Legendre symbol has useful properties, such as multiplicativity, which can s ...
34(3)
... facts reduce the two assertions above to the condition pe_u > qR+l. Divide the proof of this condition into two cases. In the first case, assume that m is the right child of m' = px ...ft_M+i#i ...9^-1. It follows from the construction of AW that p£_u+i is the largest prime dividing m' and that qR i ...
... facts reduce the two assertions above to the condition pe_u > qR+l. Divide the proof of this condition into two cases. In the first case, assume that m is the right child of m' = px ...ft_M+i#i ...9^-1. It follows from the construction of AW that p£_u+i is the largest prime dividing m' and that qR i ...
Lecture Notes - Department of Mathematics
... and described sets by axioms – fundamental properties upon which we all agree on. The most common set of axioms was formulated by Zermelo and Fraenkel. We will not discuss their axioms here, since they are more complicated than one would expect, c.f. http://en.wikipedia.org/wiki/Zermelo-Fraenkel_set ...
... and described sets by axioms – fundamental properties upon which we all agree on. The most common set of axioms was formulated by Zermelo and Fraenkel. We will not discuss their axioms here, since they are more complicated than one would expect, c.f. http://en.wikipedia.org/wiki/Zermelo-Fraenkel_set ...
The sum of divisors of n, modulo n
... We cannot prove that there are infinitely many near-perfect numbers, though we have certain Euclid-style families. For instance, if Mp := 2p − 1 is prime, then 2p−1 Mp2 is near-perfect with redundant divisor Mp . In the opposite direction, we can prove the following: ...
... We cannot prove that there are infinitely many near-perfect numbers, though we have certain Euclid-style families. For instance, if Mp := 2p − 1 is prime, then 2p−1 Mp2 is near-perfect with redundant divisor Mp . In the opposite direction, we can prove the following: ...
Elementary Number Theory Definitions and Theorems
... The definitions and results can all be found (in some form) in Strayer, but the numbering is different, and I have made some small rearrangements, for example, combining several lemmas into one proposition, demoting a “theorem” in Strayer to a “proposition”, etc. The goal in doing this was to stream ...
... The definitions and results can all be found (in some form) in Strayer, but the numbering is different, and I have made some small rearrangements, for example, combining several lemmas into one proposition, demoting a “theorem” in Strayer to a “proposition”, etc. The goal in doing this was to stream ...
11 Division Mod n - Cargal Math Books
... expressions of the from ax + by, or ax + by + cz where a, b and c are fixed integers (for example, 2x + 7y or 5x + 10y + 5z) we always get all multiples of some integer (which were defined earlier as modules). For example if we look at all values of 5x + 10y + 5z, we get 5Z which is {...!10,!5,0,5,1 ...
... expressions of the from ax + by, or ax + by + cz where a, b and c are fixed integers (for example, 2x + 7y or 5x + 10y + 5z) we always get all multiples of some integer (which were defined earlier as modules). For example if we look at all values of 5x + 10y + 5z, we get 5Z which is {...!10,!5,0,5,1 ...
Farey Sequences, Ford Circles and Pick`s Theorem
... B is the number of reticular points on the edges of the polygon. This theorem can easily be seen on a geoboard. This theorem was largely ignored until 1969 until Hugo Steinhaus included the theorem in his famous book, Mathematical Snapshots. From that point on, Pick’s Theorem has been recognized for ...
... B is the number of reticular points on the edges of the polygon. This theorem can easily be seen on a geoboard. This theorem was largely ignored until 1969 until Hugo Steinhaus included the theorem in his famous book, Mathematical Snapshots. From that point on, Pick’s Theorem has been recognized for ...
About the cover: Sophie Germain and a problem in number theory
... Germain continued to work on number theory until 1819, well after those cataclysmic events. She wrote regularly to Legendre about her efforts to develop a grand plan to prove Fermat’s Last Theorem (FLT), proving along the way an important special case today called Germain’s Theorem. Legendre was at t ...
... Germain continued to work on number theory until 1819, well after those cataclysmic events. She wrote regularly to Legendre about her efforts to develop a grand plan to prove Fermat’s Last Theorem (FLT), proving along the way an important special case today called Germain’s Theorem. Legendre was at t ...
Induction and Recursive Definition
... that one domino falling down will push over the next domino in the line. So dominos will start to fall from the beginning all the way down the line. This process continues forever, because the line is infinitely long. However, if you focus on any specific domino, it falls after some specific finite ...
... that one domino falling down will push over the next domino in the line. So dominos will start to fall from the beginning all the way down the line. This process continues forever, because the line is infinitely long. However, if you focus on any specific domino, it falls after some specific finite ...
31(2)
... the right in the n odd cases of Figure 1 and on the left in the n even cases. Other trees can be obtained by appending an arbitrary number of degree-one neighbors to the degree-two nodes in either of the trees in Figure 1. This process produces all trees Tfor which t(T)= . The upper bound also is ac ...
... the right in the n odd cases of Figure 1 and on the left in the n even cases. Other trees can be obtained by appending an arbitrary number of degree-one neighbors to the degree-two nodes in either of the trees in Figure 1. This process produces all trees Tfor which t(T)= . The upper bound also is ac ...
Prime Numbers - KSU Web Home
... actually not as good as the estimate for n = 15; 485; 863. Nonetheless, we know that as n ! 1, the value of (n) =n approaches 0 at the “same rate” that 1= ln (n) approaches 0. We are guaranteed by the Prime Number Theorem that we can make the value of ...
... actually not as good as the estimate for n = 15; 485; 863. Nonetheless, we know that as n ! 1, the value of (n) =n approaches 0 at the “same rate” that 1= ln (n) approaches 0. We are guaranteed by the Prime Number Theorem that we can make the value of ...
The Delta-Trigonometric Method using the Single
... The optimal asymptotic convergence rates are also achieved for elliptic equations of other orders. For more details, see Arnold and Wendland [3–5], Saranen and Wendland [23], Prössdorf and Schmidt [19, 20], Prössdorf and Rathsfeld [17, 18], and Schmidt [24]. Spline-spline Galerkin methods obtain ...
... The optimal asymptotic convergence rates are also achieved for elliptic equations of other orders. For more details, see Arnold and Wendland [3–5], Saranen and Wendland [23], Prössdorf and Schmidt [19, 20], Prössdorf and Rathsfeld [17, 18], and Schmidt [24]. Spline-spline Galerkin methods obtain ...