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a. Show that x² + 2y² - 8z = 5 has no integral solutions. b. Use theory to determine if x² + 14x − 35 = 0 (mod 337) has a solution. (You don't have to find this solution in case if it exists. Demonstrating the solution won't be considered as an appropriate way in the context of the question, although there could be nothing better than to have a ready solution!)

a. Show that x² + 2y² - 8z = 5 has no integral solutions. b. Use theory to determine if x² + 14x − 35 = 0 (mod 337) has a solution. (You don't have to find this solution in case if it exists. Demonstrating the solution won't be considered as an appropriate way in the context of the question, although there could be nothing better than to have a ready solution!)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter7: Real And Complex Numbers
Section7.3: De Moivre’s Theorem And Roots Of Complex Numbers
Problem 9E
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a. Show that x² + 2y² – 8z = 5 has no integral solutions. b. Use theory to determine if x² + 14x − 35 = 0 (mod 337) has a solution. (You don't have to find this solution in case if it exists. Demonstrating the solution won't be considered as an appropriate way in the context of the question, although there could be nothing better than to have a ready solution!)
Transcribed Image Text:a. Show that x² + 2y² – 8z = 5 has no integral solutions. b. Use theory to determine if x² + 14x − 35 = 0 (mod 337) has a solution. (You don't have to find this solution in case if it exists. Demonstrating the solution won't be considered as an appropriate way in the context of the question, although there could be nothing better than to have a ready solution!)
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