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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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.
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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