Number Theory Begins - Princeton University Press
... This will become a recurring theme as we continue our study of number theory. Just as we did with square numbers we will assign various traits to numbers and speak of there being prime numbers, regular numbers, perfect numbers, triangular numbers, Fibonacci numbers, Mersenne numbers—the list goes on ...
... This will become a recurring theme as we continue our study of number theory. Just as we did with square numbers we will assign various traits to numbers and speak of there being prime numbers, regular numbers, perfect numbers, triangular numbers, Fibonacci numbers, Mersenne numbers—the list goes on ...
ON THE BITS COUNTING FUNCTION OF REAL NUMBERS 1
... Remarks. The lower bound in (i) is a small improvement on Theorem 5.2 in [5] (proved with 1 + log(ad ) instead of our B(ad )). It is presented here in order to illustrate the usefulness of the simple bounds in Theorem 1. The most favorable case in (ii) is when P (X)Q(X)−aQ(X)−b is the minimal polyno ...
... Remarks. The lower bound in (i) is a small improvement on Theorem 5.2 in [5] (proved with 1 + log(ad ) instead of our B(ad )). It is presented here in order to illustrate the usefulness of the simple bounds in Theorem 1. The most favorable case in (ii) is when P (X)Q(X)−aQ(X)−b is the minimal polyno ...
Fulltext PDF - Indian Academy of Sciences
... very di®erent, but leads to the same impossibility problem as above. Recall the problem stated with reference to Figure 1. Recall that the rule is to walk in a straight line to some point of the middle vertical line as in the ¯gure and, on reaching that point, walk towards the opposite corner along ...
... very di®erent, but leads to the same impossibility problem as above. Recall the problem stated with reference to Figure 1. Recall that the rule is to walk in a straight line to some point of the middle vertical line as in the ¯gure and, on reaching that point, walk towards the opposite corner along ...
RELATED PROBLEMS 663
... 1. Ramanujan [l, p. 327, Problem 464] observed that the equation 2>w-2_7 _x2 has rational integral solutions for n and x when n = l, 2, 3, 5, and 13; and he conjectured that these were the only solutions. K. J. San j ana and T. P. Trivedi [2] discussed but did not resolve the conjecture. By means of ...
... 1. Ramanujan [l, p. 327, Problem 464] observed that the equation 2>w-2_7 _x2 has rational integral solutions for n and x when n = l, 2, 3, 5, and 13; and he conjectured that these were the only solutions. K. J. San j ana and T. P. Trivedi [2] discussed but did not resolve the conjecture. By means of ...
This paper is concerned with the approximation of real irrational
... to obtain quickly very good approximations, one would like this subsequence to be as “sparse” as possible; in this section we study the fastest way of running through the regular convergents. In doing this we pay special attention to those SRCF’s for which SRCF(x) c RCF(x), since only these can hav ...
... to obtain quickly very good approximations, one would like this subsequence to be as “sparse” as possible; in this section we study the fastest way of running through the regular convergents. In doing this we pay special attention to those SRCF’s for which SRCF(x) c RCF(x), since only these can hav ...
Week 10
... Note: The sum simplifies to 25n = 1000 because for each positive integer 1 to 12 in the sum, the corresponding integer of opposite sign −1 to −12 also appears. Then (1 + 2 + · · · + 11 + 12) + (−1 − 2 − · · · − 11 − 12) = 0. Solution 3 In this problem, we want to express 1000 as the sum of 25 consec ...
... Note: The sum simplifies to 25n = 1000 because for each positive integer 1 to 12 in the sum, the corresponding integer of opposite sign −1 to −12 also appears. Then (1 + 2 + · · · + 11 + 12) + (−1 − 2 − · · · − 11 − 12) = 0. Solution 3 In this problem, we want to express 1000 as the sum of 25 consec ...
Structure and randomness in the prime numbers
... multiplication: Fundamental theorem of arithmetic: (Euclid, ≈ 300BCE) Every natural number larger than 1 can be expressed as a product of one or more primes. This product is unique up to rearrangement. For instance, 50 can be expressed as 2 × 5 × 5 (or 5 × 5 × 2, etc.). [It is because of this theore ...
... multiplication: Fundamental theorem of arithmetic: (Euclid, ≈ 300BCE) Every natural number larger than 1 can be expressed as a product of one or more primes. This product is unique up to rearrangement. For instance, 50 can be expressed as 2 × 5 × 5 (or 5 × 5 × 2, etc.). [It is because of this theore ...
The Probability that a Random - American Mathematical Society
... Every odd prime must pass this test. Moreover, Monier [3] and Rabin [4] have shown that if n > 1 is an odd composite, then the probability that it is a strong probable prime to a random base 6, 1 < 6 < n — 1, is less than 4. Let Pi{x) denote the same probability as P(x), except that (iii) is changed ...
... Every odd prime must pass this test. Moreover, Monier [3] and Rabin [4] have shown that if n > 1 is an odd composite, then the probability that it is a strong probable prime to a random base 6, 1 < 6 < n — 1, is less than 4. Let Pi{x) denote the same probability as P(x), except that (iii) is changed ...
Products of consecutive Integers
... 4.1. If x 6 k all integers between x + k and 12 (x + k) are factors of the left hand side of the equation. According to Bertrand postulate one of them is a prime number p. Obviously that the left hand side is not divisible by p2 . Therefore it could not be an m-th power. 4.2. According to Sylvester ...
... 4.1. If x 6 k all integers between x + k and 12 (x + k) are factors of the left hand side of the equation. According to Bertrand postulate one of them is a prime number p. Obviously that the left hand side is not divisible by p2 . Therefore it could not be an m-th power. 4.2. According to Sylvester ...
THE RAMSEY NUMBERS OF LARGE CYCLES VERSUS SMALL
... If e = {u, v} ∈ E (in short, e = uv), then u is called adjacent to v, and u and v are called neighbors. For x ∈ V and a subgraph B of G, define NB (x) = {y ∈ V (B) : xy ∈ E} and NB [x] = NB (x) ∪ {x}. The degree d(x) of a vertex x is |NG (x)|; δ(G) denotes the minimum degree in G. A cycle Cn of leng ...
... If e = {u, v} ∈ E (in short, e = uv), then u is called adjacent to v, and u and v are called neighbors. For x ∈ V and a subgraph B of G, define NB (x) = {y ∈ V (B) : xy ∈ E} and NB [x] = NB (x) ∪ {x}. The degree d(x) of a vertex x is |NG (x)|; δ(G) denotes the minimum degree in G. A cycle Cn of leng ...
euler and the partial sums of the prime
... Euler knew well that nx n1 ⇡ log x (indeed, his eponymous constant measures the limiting error in this approximation [1]). So when Euler claims that “the sum of the reciprocals of the prime numbers” is “as the logarithm” of the sum of the ...
... Euler knew well that nx n1 ⇡ log x (indeed, his eponymous constant measures the limiting error in this approximation [1]). So when Euler claims that “the sum of the reciprocals of the prime numbers” is “as the logarithm” of the sum of the ...
Document
... • A natural number n is prime iff the only natural numbers dividing n are 1 and n. • The following are prime: 2, 3, 5, 7, 11, 13, … and so are 1299709, 15485863, 22801763489, … • There is an infinite number of prime numbers. • Is 2101-1=2535301200456458802993406410751 prime? • How do we check whethe ...
... • A natural number n is prime iff the only natural numbers dividing n are 1 and n. • The following are prime: 2, 3, 5, 7, 11, 13, … and so are 1299709, 15485863, 22801763489, … • There is an infinite number of prime numbers. • Is 2101-1=2535301200456458802993406410751 prime? • How do we check whethe ...
5.1. Primes, Composites, and Tests for Divisibility Definition. A
... We see that each of the four cases above give the same prime factorizasion. This is an example of the following theorem: Theorem (Fundamental Theorem of Arithmetic). Each composite number can be expressed as the product of primes in exactly one way (except for the order of the factors). Example. ...
... We see that each of the four cases above give the same prime factorizasion. This is an example of the following theorem: Theorem (Fundamental Theorem of Arithmetic). Each composite number can be expressed as the product of primes in exactly one way (except for the order of the factors). Example. ...
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"".