Are monochromatic Pythagorean triples avoidable?
... We shall denote by N the set of nonnegative integers, by N+ = {n ∈ N | n ≥ 1} the subset of positive integers, and by P = {2, 3, 5, 7, 11, 13, . . .} the subset of prime numbers. Given positive integers a ≤ b, we shall denote the integer interval they bound by [a, b] = {c ∈ N | a ≤ c ≤ b}. Definitio ...
... We shall denote by N the set of nonnegative integers, by N+ = {n ∈ N | n ≥ 1} the subset of positive integers, and by P = {2, 3, 5, 7, 11, 13, . . .} the subset of prime numbers. Given positive integers a ≤ b, we shall denote the integer interval they bound by [a, b] = {c ∈ N | a ≤ c ≤ b}. Definitio ...
Distribution of Prime Numbers 6CCM320A / CM320X
... In this chapter we discuss divisibility of integers. The most important result is the fundamental theorem of arithmetic: every integer greater than 1 can be expressed as a product of prime numbers and apart from the order of the factors this product representation is unique. We also look at some com ...
... In this chapter we discuss divisibility of integers. The most important result is the fundamental theorem of arithmetic: every integer greater than 1 can be expressed as a product of prime numbers and apart from the order of the factors this product representation is unique. We also look at some com ...
New Generalized Cyclotomy and Its Applications
... The following Generalized Chinese Remainder Theorem is frequently used in the sequel. For a proof of the Generalized Chinese Remainder Theorem and a comprehensive treatment of applications of the Chinese Remainder Theorem in computing, coding, and cryptography, we refer to [8]. LEMMA 1. ¸et m , 2 , ...
... The following Generalized Chinese Remainder Theorem is frequently used in the sequel. For a proof of the Generalized Chinese Remainder Theorem and a comprehensive treatment of applications of the Chinese Remainder Theorem in computing, coding, and cryptography, we refer to [8]. LEMMA 1. ¸et m , 2 , ...
- ScholarWorks@GVSU
... that every time we add two odd integers, the sum is an even integer. However, it is not possible to test every pair of odd integers, and so we can only say that the conjecture appears to be true. (We will prove that this statement is true in the next section.) Use of prior knowledge. This also is ...
... that every time we add two odd integers, the sum is an even integer. However, it is not possible to test every pair of odd integers, and so we can only say that the conjecture appears to be true. (We will prove that this statement is true in the next section.) Use of prior knowledge. This also is ...
Primes of the form x2 + ny2
... The first chapter will be an introduction to the problem based on the work of Fermat, who was the first known to mention results of writing primes as the sum of two squares. We will then prove his results with methods similar to those Euler used, by working out special cases of quadratic reciprocity ...
... The first chapter will be an introduction to the problem based on the work of Fermat, who was the first known to mention results of writing primes as the sum of two squares. We will then prove his results with methods similar to those Euler used, by working out special cases of quadratic reciprocity ...
Mathematical induction Elad Aigner-Horev
... Theorem 2.2. (Strong/Complete mathematical induction) Let S(n) denote a mathematical statement that depends on n ∈ Z+ . In addition, let n0 , n1 ∈ Z+ satisfy n0 ≤ n1 . (Base) If S(n0 ), S(n0 + 1), . . . , S(n1 ) are all true; and (Step) if whenever S(n0 ), S(n0 + 1), . . . , S(k − 1), S(k) are true ...
... Theorem 2.2. (Strong/Complete mathematical induction) Let S(n) denote a mathematical statement that depends on n ∈ Z+ . In addition, let n0 , n1 ∈ Z+ satisfy n0 ≤ n1 . (Base) If S(n0 ), S(n0 + 1), . . . , S(n1 ) are all true; and (Step) if whenever S(n0 ), S(n0 + 1), . . . , S(k − 1), S(k) are true ...
Farey Sequences, Ford Circles and Pick`s Theorem
... lines Pick calls a reticular polygon. Pick’s theorem states that the area of a reticular polygon is L + B/2-1 where L is the number of reticular points bordering the polygon and B is the number of reticular points on the edges of the polygon. This theorem can easily be seen on a geoboard. This theor ...
... lines Pick calls a reticular polygon. Pick’s theorem states that the area of a reticular polygon is L + B/2-1 where L is the number of reticular points bordering the polygon and B is the number of reticular points on the edges of the polygon. This theorem can easily be seen on a geoboard. This theor ...
Math 13 — An Introduction to Abstract Mathematics
... if 75% of people think a cream helps, then it probably is doing something beneficial. In mathematics and philosophy, we think very differently: the concepts of true and false and of proof are very precise. So how do mathematicians reach this blissful state where everything is either right or wrong a ...
... if 75% of people think a cream helps, then it probably is doing something beneficial. In mathematics and philosophy, we think very differently: the concepts of true and false and of proof are very precise. So how do mathematicians reach this blissful state where everything is either right or wrong a ...
Prop. If n is an integer, then 3 | (n 3 − n). Proof. By the Division
... for all integers n. This would be correct, and a uniform proof (for all primes p) follows from Fermat’s Little Theorem. Remark. You may be tempted to conjecture the following. If a is not a prime number, then a - (na − n) for some integers n. You could check this conjecture for all composite numbers ...
... for all integers n. This would be correct, and a uniform proof (for all primes p) follows from Fermat’s Little Theorem. Remark. You may be tempted to conjecture the following. If a is not a prime number, then a - (na − n) for some integers n. You could check this conjecture for all composite numbers ...
Number Theory
... (3) The greatest common factor of two integers a and b is the largest among the common factors of a and b. The greatest common factor of a and b is denoted by gcf(a, b). Remark 1.9. In the literature it is much more common to say f divides a (or f divides a evenly) than to say that f is a factor of ...
... (3) The greatest common factor of two integers a and b is the largest among the common factors of a and b. The greatest common factor of a and b is denoted by gcf(a, b). Remark 1.9. In the literature it is much more common to say f divides a (or f divides a evenly) than to say that f is a factor of ...
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"".