Read full issue - Canadian Mathematical Society
... 2. Three given circles pass through a common point P and have the same radius. Their other points of pairwise intersections are A, B , C . The 3 circles are contained in the triangle A0 B 0 C 0 in such a way that each side of 4A0 B0 C 0 is tangent to two of the circles. Prove that the area of 4A0 B0 ...
... 2. Three given circles pass through a common point P and have the same radius. Their other points of pairwise intersections are A, B , C . The 3 circles are contained in the triangle A0 B 0 C 0 in such a way that each side of 4A0 B0 C 0 is tangent to two of the circles. Prove that the area of 4A0 B0 ...
ON LARGE RATIONAL SOLUTIONS OF CUBIC THUE EQUATIONS
... by using pen and paper alone! Brouncker and Wallis were intrigued. Over the next year or so, they exchanged letters with Fermat to work out a systematic theory. They eventually found that Given a positive integer d that is not a square, one can always find infinitely many positive integer solutions ...
... by using pen and paper alone! Brouncker and Wallis were intrigued. Over the next year or so, they exchanged letters with Fermat to work out a systematic theory. They eventually found that Given a positive integer d that is not a square, one can always find infinitely many positive integer solutions ...
CHAP02 Linear Congruences
... from his celebrated “Last Theorem”. Fermat's Last Theorem states that for all integers n ≥ 3 there are no solutions to the equation xn + yn = zn for non-zero integers x, y and z. We all know that 32 + 42 = 52 and 52 + 122 = 132. There infinitely many such integer solutions to the equation x2 + y2 = ...
... from his celebrated “Last Theorem”. Fermat's Last Theorem states that for all integers n ≥ 3 there are no solutions to the equation xn + yn = zn for non-zero integers x, y and z. We all know that 32 + 42 = 52 and 52 + 122 = 132. There infinitely many such integer solutions to the equation x2 + y2 = ...
Modular Arithmetic Basics (1) The “floor” function is defined by the
... x mod m := x − m · bx/mc if m 6= 0. For positive integers x and m, x mod m = the remainder in integer division of x by m. Examples: 110 mod 26 = 6; −52 mod 26 = 0. (4) The “mod” relation is defined as follows: a ≡ b (mod m) if and only if a mod m = b mod m. The above definitions make sense even for ...
... x mod m := x − m · bx/mc if m 6= 0. For positive integers x and m, x mod m = the remainder in integer division of x by m. Examples: 110 mod 26 = 6; −52 mod 26 = 0. (4) The “mod” relation is defined as follows: a ≡ b (mod m) if and only if a mod m = b mod m. The above definitions make sense even for ...
The Abundancy Index of Divisors of Odd Perfect Numbers
... is an integer (because gcd(qi αi , σ(qi αi )) = 1). Suppose ρi = 1. Then σ(N/qi αi ) = qi αi and σ(qi αi ) = 2N/qi αi . Since N is an odd perfect number, qi is odd, whereupon we have an odd αi by considering parity conditions from the last equation. But this means that qi is the Euler prime q, and w ...
... is an integer (because gcd(qi αi , σ(qi αi )) = 1). Suppose ρi = 1. Then σ(N/qi αi ) = qi αi and σ(qi αi ) = 2N/qi αi . Since N is an odd perfect number, qi is odd, whereupon we have an odd αi by considering parity conditions from the last equation. But this means that qi is the Euler prime q, and w ...
SOME ASYMPTOTIC FORMULAS IN THE THEORY OF NUMBERS(`)
... The factor d defined by (1.4) is the largest divisor Q*(n) of n contained in L. If T is a nonvacuous subset of L, then one may define, analogous to (1.2), Presented to the Society, March 29,1961; receivedby the editors August 21,1962. 0) This research was supported in part by the National ScienceFou ...
... The factor d defined by (1.4) is the largest divisor Q*(n) of n contained in L. If T is a nonvacuous subset of L, then one may define, analogous to (1.2), Presented to the Society, March 29,1961; receivedby the editors August 21,1962. 0) This research was supported in part by the National ScienceFou ...
Section 3 - The Open University
... proof of the implication P ⇒ Q provided that each statement is shown to be true under the assumption that P is true. In Examples 3.3 and 3.4 each statement in the sequence was deduced from the statement immediately before, but the sequence can also include statements that are deduced from one or mor ...
... proof of the implication P ⇒ Q provided that each statement is shown to be true under the assumption that P is true. In Examples 3.3 and 3.4 each statement in the sequence was deduced from the statement immediately before, but the sequence can also include statements that are deduced from one or mor ...
Logic and Mathematical Reasoning
... to accommodate that. To this end we introduce quantifiers. A quantifier is a symbol which states how many instances of the variable satisfy the sentence. Definition 1.3.1 (Quantifiers). For an open setence P (x), we have the propositions (∃x)P (x) which is true when there exists at least one x for w ...
... to accommodate that. To this end we introduce quantifiers. A quantifier is a symbol which states how many instances of the variable satisfy the sentence. Definition 1.3.1 (Quantifiers). For an open setence P (x), we have the propositions (∃x)P (x) which is true when there exists at least one x for w ...
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"".