Show that the equation a2 – 1 = 0, namely (a – 1) · (a + 1) = 0 has only two solutions mod p. Please show the proof or correctness of your analysis.
Consider (p – 1)!. Recall that each element has a unique inverse. Show that (p – 1)! = -1 (mod p). Please show proof or the correctness of your analysis.
1. Show that the number of solutions (x mod p, y mod p) to the equation x² + 1 = y2 mod p is p- p (6+1) k=0
Consider the modular equation: 9x = 15 mod 30 Find all solutions. Explain or prove the correctness of your analysis.
4. Suppose that p is a prime of the form 8k + 1 . Show that the congruence x4 has ether 0 solutions or 4 solutions. 2 (mod P)
4. Suppose that p is a prime of the form 8k + 1 . Show that the congruence x4 has ether 0 solutions or 4 solutions. 2 (mod P)
28. Show that there are 12 pairs of numbers (a1,az) with 0<aj < 4,0 <a2 <6 so that x=a1 (mod 4) x = 22 (mod 6) has a solution.
I need help with number 3 on my number theory
hw.
Exercise 1. Figure out how many solutions x2 = x (mod n) has for n = 5,6,7, and then compute how many solutions there are modulo 210. Exercise 2. (a) Find all solutions to x2 +8 = 0 (mod 11). (b) Using your answer to part (a) and Hensel's Lemma, find all solutions to x2 +8 = 0 (mod 121). Exercise 3. Solve f(x) = x3 – x2 +...
Please show work clearly. Thanks
4. Suppose you had n matrices with dimensions: ai xbi ,a2 x b2. . . . ,a,, X bn. Your goal is to determine, given two integers s and t, whether it is possible to multiply a sequence from the list of given matrices together, in any order and possibly not using all of the matrices, to end up with a matrix with dimensions s × t. For example, if the list of matrix dimensions...
*16. Show that if (a,n) = (b,n) = 1, then the equation ax + by = c(mod n) has exactly n different solutions (mod n). 17 Find 11 col..ti
Laguerre's equation ty" + (1 – 2)y' + py=0) where p is a constant, is a confluent hypergeometric equation. (a) Show that the equation has x = 0) as a regular singular point with indicial root 0; (b) Show that one solution of the equation is 1+ Ž (–1)n-P(p – 1) ...(p+1-n) n=1 (n!) (©) Show that the only solutions bounded near the origin are constant multiples of the solution in (b).
please show the recurrence formula
1) Show that zo-0 is a regular singular point for the diferenta equation Zo = 0 is a regular singular point for the differential equation 15ェy" + (7 + 15r)y, +-y = 0, x>0. Use the method of Frobenius to obtain two linearly independent series solutions about zo Find the radii of convergence for these series. Form the general solution on (0, 0o). 0.
1) Show that zo-0 is a regular singular point for the...