5/7/24 80 Points 3. 2. 1. NAME on Fero Answer at least 8 problems. There are 10 points for each problem. Find a formula that describes all primitive Pythagorean triples (a, b, c) with the property that c=a+2. Let p be a prime number such that p is not a factor of 10. Let A be some given natural number such that p is a factor of (A2 5). Show that p is congruent to either 1 or 4 modulo 5. For each congruence determine whether there is a solution, (i) x² = 10 (mod 1009). It is known 1009 is a prime number. (ii) x² + X+1 = 0 (mod 37) 4. Compute the Jacobi symbol of 172 modulo 37603. 5. For each of the following numbers m, either write m as a sum of two squares or explain why it is not possible to do so. (a) 122 (b) 123 (c) 124 (d) 130 (e) 180 6. Find a formula to describe all the points on the circle X2 Y22 7. whose coordinates are rational numbers. Note that (-1,1) lies on this circle. Consider the Pell's equation X² 11 Y² = 1. A solution is X = 10, Y = 3. Determine two more solutions in natural numbers How many solutions in natural numbers are there? Explain 8. Let P be a prime number. Suppose R is a primitive root modulo P. (i) What's the order of the unit R modulo P? (ii) What's the order of the unit R² modulo P? (iii) What's the value of the Legendre symbol of R modulo P? (iv) If p > 2 then prove that there is at least one more primitive root modulo P. Suppose that q is a prime number that is congruent to 1 modulo 4 and suppose that number p=2q+1 is also a prime number. Find the value of the Legendre symbol 2 modulo p. Moreover, prove that 2 is a primitive root modulo p.
5/7/24 80 Points 3. 2. 1. NAME on Fero Answer at least 8 problems. There are 10 points for each problem. Find a formula that describes all primitive Pythagorean triples (a, b, c) with the property that c=a+2. Let p be a prime number such that p is not a factor of 10. Let A be some given natural number such that p is a factor of (A2 5). Show that p is congruent to either 1 or 4 modulo 5. For each congruence determine whether there is a solution, (i) x² = 10 (mod 1009). It is known 1009 is a prime number. (ii) x² + X+1 = 0 (mod 37) 4. Compute the Jacobi symbol of 172 modulo 37603. 5. For each of the following numbers m, either write m as a sum of two squares or explain why it is not possible to do so. (a) 122 (b) 123 (c) 124 (d) 130 (e) 180 6. Find a formula to describe all the points on the circle X2 Y22 7. whose coordinates are rational numbers. Note that (-1,1) lies on this circle. Consider the Pell's equation X² 11 Y² = 1. A solution is X = 10, Y = 3. Determine two more solutions in natural numbers How many solutions in natural numbers are there? Explain 8. Let P be a prime number. Suppose R is a primitive root modulo P. (i) What's the order of the unit R modulo P? (ii) What's the order of the unit R² modulo P? (iii) What's the value of the Legendre symbol of R modulo P? (iv) If p > 2 then prove that there is at least one more primitive root modulo P. Suppose that q is a prime number that is congruent to 1 modulo 4 and suppose that number p=2q+1 is also a prime number. Find the value of the Legendre symbol 2 modulo p. Moreover, prove that 2 is a primitive root modulo p.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 92E
Related questions
Question
9
![5/7/24
80 Points
3.
2.
1.
NAME
on
Fero
Answer at least 8 problems. There are 10 points for each problem.
Find a formula that describes all primitive Pythagorean triples (a, b, c) with
the property that
c=a+2.
Let p be a prime number such that p is not a factor of 10. Let A be some given
natural number such that p is a factor of (A2 5). Show that p is
congruent to either 1 or 4 modulo 5.
For each congruence determine whether there is a solution,
(i)
x² =
10 (mod 1009). It is known 1009 is a prime number.
(ii)
x² + X+1 = 0 (mod 37)
4. Compute the Jacobi symbol of 172 modulo 37603.
5.
For each of the following numbers m, either write m as a sum of two squares
or explain why it is not possible to do so.
(a) 122 (b) 123 (c) 124 (d) 130 (e) 180
6. Find a formula to describe all the points on the circle
X2 Y22
7.
whose coordinates are rational numbers. Note that (-1,1) lies on this circle.
Consider the Pell's equation X²
11 Y² = 1.
A solution is X = 10, Y = 3. Determine two more solutions in natural numbers
How many solutions in natural numbers are there? Explain
8. Let P be a prime number. Suppose R is a primitive root modulo P.
(i) What's the order of the unit R modulo P?
(ii) What's the order of the unit R² modulo P?
(iii) What's the value of the Legendre symbol of R modulo P?
(iv) If p > 2 then prove that there is at least one more primitive root modulo P.
Suppose that q is a prime number that is congruent to 1 modulo 4 and suppose that
number p=2q+1 is also a prime number. Find the value of the Legendre symbol
2 modulo p. Moreover, prove that 2 is a primitive root modulo p.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F30e6ca29-61bc-41d1-bea3-28d53872fdaf%2F58c7f974-6e3c-44a3-a71e-940b8c6d73aa%2Fvd3lttg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5/7/24
80 Points
3.
2.
1.
NAME
on
Fero
Answer at least 8 problems. There are 10 points for each problem.
Find a formula that describes all primitive Pythagorean triples (a, b, c) with
the property that
c=a+2.
Let p be a prime number such that p is not a factor of 10. Let A be some given
natural number such that p is a factor of (A2 5). Show that p is
congruent to either 1 or 4 modulo 5.
For each congruence determine whether there is a solution,
(i)
x² =
10 (mod 1009). It is known 1009 is a prime number.
(ii)
x² + X+1 = 0 (mod 37)
4. Compute the Jacobi symbol of 172 modulo 37603.
5.
For each of the following numbers m, either write m as a sum of two squares
or explain why it is not possible to do so.
(a) 122 (b) 123 (c) 124 (d) 130 (e) 180
6. Find a formula to describe all the points on the circle
X2 Y22
7.
whose coordinates are rational numbers. Note that (-1,1) lies on this circle.
Consider the Pell's equation X²
11 Y² = 1.
A solution is X = 10, Y = 3. Determine two more solutions in natural numbers
How many solutions in natural numbers are there? Explain
8. Let P be a prime number. Suppose R is a primitive root modulo P.
(i) What's the order of the unit R modulo P?
(ii) What's the order of the unit R² modulo P?
(iii) What's the value of the Legendre symbol of R modulo P?
(iv) If p > 2 then prove that there is at least one more primitive root modulo P.
Suppose that q is a prime number that is congruent to 1 modulo 4 and suppose that
number p=2q+1 is also a prime number. Find the value of the Legendre symbol
2 modulo p. Moreover, prove that 2 is a primitive root modulo p.
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