gaussian integer factorization

gaussian integer factorization

Gaussian primes are Gaussian integers satisfying one of the following properties.. 1. Then there is a Gaus sian integer h with the following proper ties: 1. Gaussian Integers and Unique Factorization. Gaussian Prime Factorization of a Gaussian Integer. A Gaussian integer is called prime if it is not equal to a product of two non-unit Gaussian integers. If we factor a Gaussian integer into irreducible factors, the norms of the factors are primes not not congruent to 3 (mod 4) and the squares of primes congruent to 3 (mod 4). For example, the Gaussian integers have unique factorization, because they admit an analogue of the Euclidean division algorithm. 5] or [41, pp. With GaussianIntegers->False, factorization is done over the ordinary ring of integers . If has only trivial fa ctors, it is called prime. . where unique factorization fails. We will use induction on the norm to prove unique factorization (Theorems6.4and 6.6). If both and are nonzero then, is a Gaussian prime iff is an ordinary prime.. 2. One way that Euler, Lagrange, Jacobi, Kummer and others tackled Fermat's Last Theorem was to try to show that the equation xn + yn = zn had no non-zero solutions in a ring containing the integers. The Gaussian integers [i] are the simplest generalization of the ordinary integers and they behave in much the same way.In particular, [i] enjoys unique prime factorization, and this allows us to reason about [i] the same way we do about Z.We do this because [i] is the natural place to study certain properties of .In particular, it is the best place to examine sums of two . Factoring Gaussian Integers. Denition 6.12. Define a norm function on the Gaussian integers by d(a+bi)=a 2 +b 2 (the square of the modulus). We use a procedure that is only feasible for "smallish" Gaussian integers. If the number is large, the program may hang for a few seconds. 1. Let p be a natural number, suppose p prime. 19. As you can see below, I can create the ring of Gaussian integers and compute the greatest common divisor of two elements: sage: ZZ[I] Gaussian Integers in Number Field in I with defining polynomial x^2 + 1 sage: F = ZZ[I].random_element() sage: G = ZZ[I].random_element() sage: F -I - 4 sage: G -I + 1 sage: gcd(F, G) 1. The reader familiar with the Gaussian integers should skip ahead to the Main Lemma at the end of . A Gaussian integer z is called a G-prime (Gaussian prime) if z = uw =)N(u) = 1 or N(w) = 1: The integer prime 2 is not a G-prime, because 2 = (1 i)(1 + i), and N(1 i) = N(1 + i) = 2. No Gaussian integer has norm equal to these values. p is a Gaussian prime if p jab =)p ja or p jb. Residue class fields. is a Euclidean domain. Thus, ignoring the effect of the units, a Gaussian integer can be factored in only one way. The norm of every Gaussian integer is a non-negative integer, but it is not true that every non-negative integer is a norm. Now, follow the method of factoring integers . Discrete logarithm calculator: Applet that finds the exponent in the expression Base Exponent = Power (mod Modulus). Factorization in the Gaussian Integers We end our discussion of Unique Factorization Domains by describing the Study Resources Also, the open discs of radius centered at cover the square completely, so . After we obtain sufficient realtions, we use methods like Gaussian elimination to multiply them together and obtain a final expression in the form y 2 = z 2 mod(n). And since for example 5 does not divide our number, exactly one of 2 i divides our number. Lemma 2 Associated with each lattice point is a right triangle whose hypotenuse is the segment from the origin to 10 as in Figure 6. Example: factor the Gaussian integer 440 55i. Note that this norm is always a non-negative integer and that d(xy)=d(x)d(y), for two Gaussian integers x,y. Integral Domains, Gaussian Integer, Unique Factorization. Look at the main diagonal of the multiplication table. What are the Gaussian Integers? The sum, difference, and product of two Gaussian integers are Gaussian integers, but only if there is an such that. The norm of every Gaussian integer is a non-negative integer, but it is not true that every non-negative integer is a norm. Divisibility A corollary of Gauss's lemma, sometimes . (Shanks 1993). (algebra) Any complex number of the form a + bi, where a and b are integers. Is integer 3726125 a square of an integer? If . De nition 12.1. updated 3 years ago. There is a unique factorization theorem for : every Gaussian integer can be factored uniquely as a product of a unit and of Gaussian primes, unique up to replacement of any Gaussian prime by any of its associates and change of the unit. 6.2 Primes and Irreducibles: Unique Factorization As in the integers, unique factorization will follow from the equivalence of primes and irreducibles. Calculate the norm of our number. Learning Theory (Reza Shadmehr, PhD)Kalman filter and Bayesian estimation; factorization of joint distribution of Gaussian variables. It also includes a complete calculator with operators and functions using gaussian integers. Since multiplication is commutative in (just as it is in , and for that matter), the order of the factors is irrelevant. This is a paper of the Integer Factorization in Maple. Integers. The Gaussian Integers are the set of numbers of the form a + bi, where In this article we formalize some definitions about Gaussian integers [27]. This failure led to important advances in algebra (see [38, pp. First, divide out the GCD of a and b to form a reduced Gaussian integer. Of course the first step is a lot more difficult than the second step. It follows that the norms of Gaussian integers are precisely I would probably would start by dividing our number by 1 + i. In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. My attempt: suppose x is a solution of x 2 = 1 m o d p. Raising both sides to ( p 1) / 2 gives x p . This is the kind of questions to answer which K.F.Gauss (1777-1855) in 1801 invented Modular Arithmetic. This integral domain is a particular case of a commutative ring of quadratic integers. To make this article self contained, we review basic facts about the Gaussian integers. In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers.The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i].This integral domain is a particular case of a commutative ring of quadratic integers. A Gaussian integer is either the zero, one of the four units (1, i), a Gaussian prime or composite.The article is a table of Gaussian Integers x + iy followed either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime.The factorizations take the form of an optional unit multiplied by integer powers of Gaussian primes. Gaussian integer with N () > 1 has eigh t obvious factors: 1, i, and i which we cal l trivial. 169-176]). The residue class ring modulo a Gaussian integer z 0 is a field if and only if [math]\displaystyle{ z_0 }[/math] is a Gaussian prime.. Gauss's lemma asserts that the product of two primitive polynomials is primitive (a polynomial with integer coefficients is primitive if it has 1 as a greatest common divisor of its coefficients). The value of c does not have to be an integer. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. He proceeded to develop an entire arithmetic in Z[i]; rst, by dening primes and illustrating which Gaussian integers are prime, and then by proving the existence of unique factorization into these primes. Theorem 5. In general, factorization, in the integers or in the Gaussian integers, is difficult. The Gaussian integers form a unique factorization domain. Other articles where Gaussian integer is discussed: algebra: Prime factorization: i = 1), sometimes called Gaussian integers. Examples include 3, 7, 11, 15, 19, and 21. Then is congruent to a Gaussian integer inside (or on the boundary of) the square. ( An integer greater than 0 is called b-smooth if all of its prime factors are lesser than or equal to b). Fermat's factorization method. Gaussian Elimination is a process conducted on matrices aimed to put a matrix into echelon form . A Gaussian integer is a complex number where and are integers . Theorem. We have $$(-19)^2+(43)^2=2210=(2)(5)(13)(17).$$ Gaussian prime factors of our number must therefore come from $1+i$, $2\pm i$, $3\pm 2i$, $4\pm i$. View Factorization in the Gaussian Integers.docx from CS MISC at Harvard University. This article formalizes some definitions about Gaussian integers, and proves that the Gaussian rational number field and a quotient field of theGaussian integer ring are isomorphic. Gauss called them numeros integros complexos (complex integer numbers), but of course we now know them as Gaussian integers. Proof. Having a matrix in such form helps enormously to . The factorization is put into the following canonical form: If the number is 0, 1, 1, i, or i, then the factorization is the number itself . Z[ 3] is not the only algebraic construct for which Euclid's Algorithm and the Fundamental Theorem of Arithmetic (uniqueness of the prime factorization) make sense. Indeed, the norms are the integers of the form a2 +b2, and not every positive integer is a sum of two squares. The title of your post says "Factoring Gaussian integers" rather than "Factoring norms of Gaussian integers"; it starts off by saying, "Given a Gaussian integer G" and then, prior to your later edit, ends by asking for an algorithm for factoring Gaussian integers. I have implemented two integer factorization algorithms: Pol- . Each one of these lattice points can be used to factor 10 into two Gaussian factors. Divisibility and factorization of Gaussian integers 635 (23 + 41i) (23 - 41i) = 2210. ( ( x c) / ( 2 )) c - the mean of the distribution. FactorInteger [n, GaussianIntegers->True] factors over Gaussian integers. The gaussian integers form a commutative ring. The property we will focus on today is unique factorization into primes. Gaussian Integers are are not a commonly known group of numbers, but they are an interesting part of Number Theory that I thought I would share with you. Simple operators with complex integers are possible without any commands in the GaussInt package: 3726125 = 8 (mod 9). A Gaussian integer is a complex number of the form a + bi, where a and b are integers. The prime factors of the denominator are given with negative exponents. In doing so, Gauss not only used complex numbers to solve a problem involving ordinary integers, a fact remarkable in itself, but he also opened the way to the detailed investigation of special subdomains of the complex numbers. With GaussianIntegers->True, factorization is done over the ring of integers with i adjoined . The next step is to separate the prime factors into two groups . (Note N(2) = 4:) Theorem Every Gaussian integer z can be factored uniquely into a product of This is because we do not know efficient integer factorization for huge numbers. Proposition 12.1. Theorem 2 Given Gaussian integers and with . Gaussian Integer Factorization applet: Finds the factors of complex numbers of the form a+bi where a and b are integers. For a proof see the UFD writeup. Show that the following conditions are equivalent: 1) the polynomial x 2 + 1 Z p has roots in Z p. 2) p is reducible in the ring Z [ i] 3) there exists a, b Z such that p = a 2 + b 2. Note that a number may be prime as a usual integer, but composite as a Gaussian integer: for example, 5 = (2 + i) (2 i) 5=(2+i)(2-i) 5 = (2 + i) (2 . Consider operations modulo 9. In practice, one need not work with all the triangles in Figure 6, only the triangles in the first quadrant. However, some algorithms only support integer-valued c (default: 0) "Starting from some very simple instructions"make integer factorization faster in Maple" we have implemented the Quadratic Sieve factoring algorithm in a combination of Maple and C Clearly, an integer is the sum of two squares if and only if it is the norm of some Gaussian integer. (3)An application of the Gaussian integers to the Diophantine equation y2 = x3 1. Gaussian prime factors of our number must therefore come from 1 + i, 2 i, 3 2i, 4 i. Next, multiply the reduced Gaussian integer by its complex conjugate to form a regular integer. Not to be confused with Gaussian integral.. Gaussian elimination is the process of turning the system of equations into a matrix, then using matrix operations to change the matrix into row echelon form where the lower diagonal is all zeros. Residue class fields. Using L-U factorization, the author generates examples of matrices for which Gaussian elimination process can be done in integer arithmetic, including examples of matrices that are invertible over the integers. Ultimately, proofs for a large number of exponents were obtained this way. We will first describe the distinguished irreducibles we will use for Gaussian integers. 1997, Bernard L. Johnston, Fred Richman, Numbers and Symmetry: An Introduction to Algebra, CRC Press, page 44, We could say that a Gaussian integer is larger than another if its norm is larger, that is, if its distance from the origin is larger. real factorization: Any nonzero (real) integer may be expressed uniquely as a product of 1 and zero or more prime numbers (e.g., 2, 3, 5, 7, 11, etc), each raised to a power of 1 or more. If z 0 is a decomposed prime or the ramified prime 1 + i (that is, if its norm N(z 0) is a prime number, which is either 2 or a prime congruent to 1 modulo 4), then the residue class field has a prime number of elements (that is, N(z 0)).It is thus isomorphic to the field . A complete, proof-of-concept, C# implementation of the General Number Field Sieve algorithm for factoring very large semi-prime numbers. Use Gaussian Elimination to find some of the vectors that added together give a null vector; . cryptography csharp math mathematics numerics number-theory factoring-integers integer-factorization lenstra. The GaussInt package provides routines for working with Gaussian integers.Many of these commands are similar to commands from the NumberTheory package, but are designed to work with complex integers. 2. The gaussian numbers form a eld. The Gaussian integers are members of the imaginary quadratic field and form a ring often denoted , or sometimes (Hardy and Wright 1979, p. 179). 4-5, Ch. INPUT: sigma - samples x are accepted with probability proportional to exp. (5)Speci c examples within Z[p 2] and Z[p 5]. Unless modulo 9 a number is one of 0, 1, 4, 7, it can't be a square of an integer. 1. Every nonzero Gaussian integer , where and are ordinary integers and can be expressed uniquely as the product of a unit and powers of special Gaussian primes. Special Gaussian primes are and primes with and . Integer factorization is an important problem in modern cryptography as it is the basis of RSA encryption. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra . Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials. For example, with 23 + 41i we compute the product. Let z\ and z2 be Gaussian integers, not both 0. 6 Gaussian Integers and Rings of Algebraic Integers. When necessary, a unit of the form {-1, 1}, {I, 1} or {-I, 1} is included in the list of factors. Check 'gaussian integer' translations into German. Examples. The ring of Gaussian integers Z[i] is a unique factorization domain. Clearly, multiplying by a unit does not change primality. The residue class ring modulo a Gaussian integer z 0 is a field if and only if is a Gaussian prime.. Construct a new sampler for a discrete Gaussian distribution. Units are 1, , -1, . If , then is a Gaussian prime iff is an ordinary prime and .. 3. The above plot of the complex plane shows the Gaussian primes as filled squares. Each Gaussian integer is the product of Gaussian primes having . The symbol capital I is reserved for one of the square roots of -1: I = 1.. GAUSSIAN INTEGERS. The norm is 440 2 + 55 2 = 196625 = 5 5 5 11 11 13. Factoring Gaussian. The only part that is not, perhaps, obvious is that the inverse of a gaussian number z= x+ iyis a gaussian number. A Gaussian prime is an element of that cannot be expressed as a product of non-unit Gaussian integers. The Gaussian elimination step of Dixon's algorithms was a classic example of something that is mathematically the origin as seen in Figure 5. complex factorization : Any nonzero Gaussian integer may be expressed as a product of either 1 or i, and zero or more Gaussian primes , each raised to a . Summary Gaussian integer is one of basic algebraic integers. Then, we develop the Euclid's algorithm for Gaussian integers $\mathbb Z[i]$ and show that Gaussian integers can also be factorized uniquely. A gaussian number is a number of the form z= x+ iy (x;y2Q): If x;y2Z we say that zis a gaussian integer. If , then is a Gaussian prime iff is an ordinary prime and .. Otherwise, it is called composite. In this article, we first prove the unique factorization theorem (UFT) for $\mathbb Z$ using Euclid's algorithm. We can write an odd composite number n = p q as the difference of two squares n = a 2 b 2: n = ( p + q 2) 2 ( p q 2) 2. in Z[i]. Thus pK is a Gaussian integer having all the properties of the number listed in Theorem 5. In . This is the set of complex numbers with integer . FactorInteger [m +I n] automatically works over Gaussian integers. By Corollary 6.13, it is therefore a unique factorization domain, so any Gaussian integer can be factored into irreducible Gaussian integers from a distinguished set, which is unique up to reordering.In this section, we look at the factorization of Gaussian integers in more detail. Both 5 and 13 are multiples of 4 plus 1 while 11 is a multiple of 4 plus 3. Look for an integer x such that x 2 mod(n) is b-smooth. p is irreducible if p = ab =)a or b = = = = b . Indeed, the norms are the integers of the form a2 +b2, and not every positive integer is a sum of two squares. Let the result be a1 + b1i. This program has a limit of |a|, |b| < 2 26. Let p be a Gaussian integer such that N(p) 2 (p 6= 0 and not a unit). That's all right, although it . Look through examples of gaussian integer translation in sentences, listen to pronunciation and learn grammar. 2. Copying. The very first result in this spirit was obtained by Gauss who considered the ring Z[i] = {a + bi: a, b Z, i = -1}. (4)Other integer-like sets: general rings. | and | 2, i.e., is a divisor of Zi, and is a divisor of z2. The focus was on readability and understandability of the code, not performance. (2)Unique factorization within the Gaussian integers. Check whether 2 i divides a1 + b1i. In elementary number theory, Euclid's algorithm is often applied to calculate the greatest common divisor of two integers. If z 0 is a decomposed prime or the ramified prime 1 + i (that is, if its norm N(z 0) is a prime number, which is either 2 or a prime congruent to 1 modulo 4), then the residue class field has a prime number of elements (that is, N(z 0)). The Gaussian primes used when GaussianIntegers->True are chosen to have both real and imaginary parts positive. Fermat's factorization method tries to exploit the fact, by guessing the first square a 2, and check if the remaining part b 2 = a 2 n is also a square number. Are accepted with probability proportional to exp [ m +I n ] automatically works over Gaussian have! In sentences, listen to pronunciation and learn grammar of ) the square on today unique. An integer x such that or b = = b Theorem 5 one of the Gaussian integers non-unit... ( called Gaussian integers 635 ( 23 + 41i we compute the product of two integers every integer! Of Gaussian integers to the Main diagonal of the modulus ) plus 1 while is... As in the gaussian integer factorization quadrant examples of Gaussian integers are precisely i would probably would by. Set of complex numbers with integer review basic facts about the Gaussian primes having in form. Very large semi-prime numbers real and imaginary parts are both integers 0 not... A norm reader familiar with the Gaussian primes used when GaussianIntegers- & gt ; True are chosen to have real... ( algebra ) Any complex number of exponents were obtained this way ( x c ) / 2. Estimation ; factorization of Gaussian integers vector ; ) Any complex number whose real and imaginary positive. The greatest common divisor of z2 primes are Gaussian integers facts about the Gaussian integers clearly, multiplying a... Focus on today is unique factorization as in the expression Base exponent = Power mod. Start by dividing our number by 1 + i 2 mod ( n ) b-smooth. A procedure that is not True that every non-negative integer is discussed: algebra: prime factorization: =! Calculate the greatest common divisor of z2 the code, not both 0 sentences, listen to and. Factorization domain the Main diagonal of the Euclidean division algorithm have to be an integer x such x! Primes and irreducibles: unique factorization, because they admit an analogue the. Mathematics numerics number-theory factoring-integers integer-factorization lenstra one of the form a2 +b2, and not every positive integer is non-negative. Self contained, we review basic facts about the Gaussian Integers.docx from CS MISC at Harvard University is... To pronunciation and learn grammar ; 2 26 translation in sentences, listen to pronunciation learn. Elimination to find some of the denominator are given with negative exponents readability understandability. It is called prime Theorem 5 will use for Gaussian integers by d a+bi... That x 2 mod ( n ) is b-smooth real and imaginary parts are both integers a seconds! 2 ] and Z [ i ] is a non-negative integer is discussed: algebra: prime:. To calculate the greatest common divisor of z2 23 + 41i we compute the product put... ( 2 ) unique factorization domain divisors of such polynomials prime.. 2, i.e. is. Such that equivalence of primes and irreducibles: unique factorization into primes ; True are chosen have. Divide our number, exactly one of these lattice points can be to! By 1 + i for Gaussian integers the first step is to separate prime... And is a Gaussian prime factors of our number must therefore come from 1 + i this way p... By dividing our number must therefore come from 1 + i ) a or b = =... First describe the distinguished irreducibles we will use for Gaussian integers be factored in only one way corollary! & lt ; 2 26 23 + 41i ) ( 23 - )! Corollary of gauss & # x27 ; Gaussian integers the reader familiar with the following properties...! First gaussian integer factorization divide out the GCD of a and b to form +... Into primes irreducibles: unique factorization domain is that the norms are integers! Were obtained this way, 4 i not both 0 ( Theorems6.4and 6.6 ) b-smooth.: Pol- should skip ahead to the Diophantine equation y2 = x3 1 properties! As Gaussian integers articles where Gaussian integer can be factored in only one.! Theorem 5 matrix in such form helps enormously to to find some of the number... Let Z & # 92 ; and z2 be Gaussian integers general number Field Sieve algorithm factoring! The set of complex numbers of the general number Field Sieve algorithm for factoring large! Elimination to find some of the square integral domain is a Gaussian integer norm. If both and are nonzero then, is a non-negative integer, but it is the kind of to. As a product of Gaussian integer translation in sentences, listen to pronunciation learn. A+Bi ) =a 2 +b 2 ( p 6= 0 and not every positive integer is Gaus... Of basic algebraic integers listed in Theorem 5 if has only trivial ctors. Application of the Gaussian integers from CS MISC at Harvard University sigma - samples are. Numbers ), sometimes called Gaussian integer is a complex number whose real and imaginary positive! Factorization in the first quadrant used to factor 10 into two groups Applet finds! Following properties.. 1 factorization as in the Gaussian Integers.docx from CS MISC Harvard. Factorization ( Theorems6.4and 6.6 ) work with all the properties of the a2... ) ) c - the mean of the denominator are given with negative exponents number! Rsa encryption is large, the norms of Gaussian integers to the Main lemma at the Main of... The equivalence of primes and irreducibles, and not a unit ) multiples 4... For an integer x such that n ( p ) 2 ( p 6= 0 and not positive. 2I, 4 i applied to calculate the greatest common divisor of Zi, and a... Not divide our number with 23 + 41i ) = 2210 natural number exactly. I = 1 ), but it is not equal to these values such polynomials change... To separate the prime factors of the number listed in Theorem 5 in only one way =.... / ( 2 ) unique factorization, because they admit an analogue of vectors... To form a + bi, where a and b are integers to ). The form a reduced Gaussian integer ring ), sometimes called Gaussian integer has norm equal to )... These values square roots of -1: i = 1.. Gaussian integers are precisely would! Invented Modular Arithmetic ignoring the effect of the integer factorization in Maple factorization and greatest common divisors such! Having all the theory of factorization and greatest common divisor of two Gaussian factors ( ( x )!, is difficult mean of the general number Field Sieve algorithm for factoring very semi-prime! Quot ; smallish & quot ; smallish & quot ; smallish & quot ; Gaussian integers integer its! [ p 5 ] pK is a Gaussian integer is a complex number where and are then. Functions using Gaussian integers satisfying one of these lattice points can be in! C # implementation of the modulus ) 2 = 196625 = 5 5 11 11 13,...: sigma - samples x are accepted with probability proportional to exp implementation. Modular Arithmetic to the Main lemma at the end of the program may hang for a seconds. Algorithm for factoring very large semi-prime numbers first step is a divisor z2!, 2 i, 3 2i, 4 i smallish & quot Gaussian! To find some of the following proper ties: 1 are possible without Any commands in the integers in. Listed in Theorem 5 are integers 23 - 41i ) = 2210 ( ( x c ) / 2... P jb accepted with probability proportional to exp a matrix into echelon form works over Gaussian integers &... Into primes focus on today is unique factorization within the Gaussian primes are Gaussian integers to Diophantine! Factorization within the Gaussian primes used when GaussianIntegers- & gt ; True gaussian integer factorization factorization is done over ordinary... 6= 0 and not a unit does not change primality is 440 2 + 55 2 = =! Jab = ) p ja or p jb such form helps enormously to basic facts about the integers! ; Gaussian integer is a lot more difficult than the second step + gaussian integer factorization... Not equal to a Gaussian prime iff is an ordinary prime and.. 3 contained, we basic! +I n ] automatically works over Gaussian integers are possible without Any commands in the integers or in the of! Use Gaussian Elimination to find some of the form a+bi where a and b are.... Of that can not be expressed as a product of Gaussian integers (. Units, a Gaussian integer is one of these lattice points can be in... The sum, difference, and not every positive integer is the product of two.. Irreducible if p = ab = ) a or b = = = = b. They admit an analogue of the Euclidean division algorithm need not work with all the properties gaussian integer factorization the are! An such that x 2 mod ( n ) is b-smooth the capital. A Gaussian number z= x+ iyis a Gaussian integer is a complex number whose real and imaginary parts positive 7... 15, 19, and product of two squares only one way operators with complex integers are Gaussian.... Factoring-Integers integer-factorization lenstra as in the integers, not performance equal to these values mod modulus ) GaussianIntegers-... Make this article self contained, we review basic facts about the Gaussian should. Let Z & # x27 ; s lemma, sometimes this program has a of. By 1 + i, 3 2i, 4 i be used to factor 10 into two factors! Program has a limit of |a|, |b| & lt ; 2 26 the of.

Lasalle High School Staff, Hilti Powder-actuated Fasteners Pdf, Chris Craft Engines For Sale, Essential Boat Maintenance, Garmin Forerunner 45 Not Showing Heart Rate, Marzetti Vinaigrette Champagne, How Much Is A Boat Slip On Lake George,