Prime Portraits - The Bridges Archive
... Marie-Sophie Germain for her work relating to Fermat’s Last Theorem) are those prime numbers p for which 2p + 1 is also prime, such as 53 (because 2 · 53 + 1 = 107 is also prime). The 53 × 107-digit prime number Psg in Figure 5 is not merely a punny “Sophie Germain” prime by virtue of depicting her; ...
... Marie-Sophie Germain for her work relating to Fermat’s Last Theorem) are those prime numbers p for which 2p + 1 is also prime, such as 53 (because 2 · 53 + 1 = 107 is also prime). The 53 × 107-digit prime number Psg in Figure 5 is not merely a punny “Sophie Germain” prime by virtue of depicting her; ...
(i) Suppose that n > 1 is a composite integer, with n = rs, say. Show
... (4) Suppose that n pairs of gloves of different sizes are mixed together in a drawer. How many individual gloves must you take out if you are to be sure of having at least one complete pair. (Of course you must justify your answer!) Solution: There are n pairs of gloves. Imagine that each individual ...
... (4) Suppose that n pairs of gloves of different sizes are mixed together in a drawer. How many individual gloves must you take out if you are to be sure of having at least one complete pair. (Of course you must justify your answer!) Solution: There are n pairs of gloves. Imagine that each individual ...
solutions.
... number with exactly two factors (or one repeated factor if the prime was multiplied by itself—but this still makes it composite): the two primes that were multiplied together. A nonprime multiplied by a nonprime never results in a prime, essentially for the same reason as above. Note: the number 1 i ...
... number with exactly two factors (or one repeated factor if the prime was multiplied by itself—but this still makes it composite): the two primes that were multiplied together. A nonprime multiplied by a nonprime never results in a prime, essentially for the same reason as above. Note: the number 1 i ...
Solutions
... Solution: a) A function f : A −→ B is surjective (or onto) if for any b ∈ B there is a ∈ A such that f (a) = b (so B is equal to the range of f ). We say that the function g : B −→ A is the inverse function of f : A −→ B if f ◦ g = idB (i.e. f (g(b)) = b for all b ∈ B) and g ◦ f = idA (i.e. g(f (a)) ...
... Solution: a) A function f : A −→ B is surjective (or onto) if for any b ∈ B there is a ∈ A such that f (a) = b (so B is equal to the range of f ). We say that the function g : B −→ A is the inverse function of f : A −→ B if f ◦ g = idB (i.e. f (g(b)) = b for all b ∈ B) and g ◦ f = idA (i.e. g(f (a)) ...
Sample pages 1 PDF
... [4] P. R IBENBOIM : The New Book of Prime Number Records, Springer-Verlag, ...
... [4] P. R IBENBOIM : The New Book of Prime Number Records, Springer-Verlag, ...
doc - WHRO Education
... guaranteed is that chair number 1 is skipped on the very first circuit. Sample: if N = 5 then the chairs would be 1, 2, 3, 4, 5 but after the first time around the prisoners in chairs 2 and 4 are dead leaving only 1 3 5, but since 5 was skipped poor number 1 is next to go, 3 is skipped and finally 5 ...
... guaranteed is that chair number 1 is skipped on the very first circuit. Sample: if N = 5 then the chairs would be 1, 2, 3, 4, 5 but after the first time around the prisoners in chairs 2 and 4 are dead leaving only 1 3 5, but since 5 was skipped poor number 1 is next to go, 3 is skipped and finally 5 ...
PRIME NUMBERS 1. Prime Divisors Theorem 1. If n > 1 is
... Definition 4. A number of the form 22 + 1 is called a Fermat number. A prime that is a Fermat number is called a Fermat prime. The first five Fermat numbers are 3, 5, 17, 257, and 65537. Fermat checked that all such numbers up to 216 + 1 = 65537 are prime, and conjectured that all of them are prime. ...
... Definition 4. A number of the form 22 + 1 is called a Fermat number. A prime that is a Fermat number is called a Fermat prime. The first five Fermat numbers are 3, 5, 17, 257, and 65537. Fermat checked that all such numbers up to 216 + 1 = 65537 are prime, and conjectured that all of them are prime. ...
The distribution of quadratic and higher residues, (1)
... of times, we say that it is of the first type. In this case, the polynomial (x+ n,). . .(x+ n,,) is a perfect square, and the value of the sum extended over x lies between p- r and p. Hence the contribution of the sets of integers n1, ---, n,,. of the first type is w-, h) (P-W, where F(r, h) is the ...
... of times, we say that it is of the first type. In this case, the polynomial (x+ n,). . .(x+ n,,) is a perfect square, and the value of the sum extended over x lies between p- r and p. Hence the contribution of the sets of integers n1, ---, n,,. of the first type is w-, h) (P-W, where F(r, h) is the ...
On the prime factors of the number 2 p-1 - 1
... (Received 6 April, 1967) From the proof of Theorem 2 of [5] it follows that for every positive integer k there exist infinitely many primes/? in the arithmetical progression ax+b (x = 0, 1,2,...), where a and b are relatively prime positive integers, such that the number 2 P ~ 1 — 1 has at least k c ...
... (Received 6 April, 1967) From the proof of Theorem 2 of [5] it follows that for every positive integer k there exist infinitely many primes/? in the arithmetical progression ax+b (x = 0, 1,2,...), where a and b are relatively prime positive integers, such that the number 2 P ~ 1 — 1 has at least k c ...
