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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
4
![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%2Fcdc5b0c6-0441-485f-b330-2290b153d998%2Fth4qyog_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.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)