Evaluate the Legendre symbol () (a) using the reciprocity law 1873 8389 only for the Legendre symbol and (b) using the reciprocity law for the Jacobi symbol.
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- Suppose you draw an upside down staircase with boxes that create an x by x grid. The uncolored staircase attached photo shows heights 2, 3, and 4. Tiling a staircase of height, x, means to create nonoverlapping rectangles within the staircase. Prove that tiling with x rectangles is a Catalan object. Note: A Catalan object is proven by one of the following (i) a bijection of a known Catalan object (ii) show it satifies a Catalan recursion (iii) directly count it and find there are (1/(x+1))*(2x choose x) of them Tilings of a height three staircase with three rectangles of different colors are pictured in the attached photo.Use Cauchey's Residue TheoremPlease show full and clear steps in workingCan you please provide a bit more details to the first part of the proof where L=SupR. How do you know that sup R is a lower bound for S? How do you know that it is greater than or equal to every other lower bound?
- Suppose you draw an upside down staircase with boxes that create an x by x grid. The uncolored staircase attached photo shows heights 2, 3, and 4. Tiling a staircase of height, x, means to create nonoverlapping rectangles within the staircase. Prove that tiling with x rectangles is a Catalan object. Note: A Catalan object is proven by one of the following (i) a bijection of a known Catalan object (ii) show it satifies a Catalan recursion Tilings of a height three staircase with three rectangles of different colors are pictured in the attached photo.Consider the equivalence rule 7.2.9c. Note that x does not occur free in C in this equivalence.Prove 7.2.9c using other equivalences. That is, start with the left-hand-side of 7.2.9c, and use otherequivalences to transform the left hand side to equivalent formulas until you obtain the right-handside of 7.2.9c (or start with the right-hand-side and obtain the left-hand-side). Write down yourwork step by step very clearly, stating which equivalence you are using for each step.Working directly from definition, prove that if zn and wn are sequences of complex numbers withlimn→∞zn = 4 + 3i, and limn→∞wn = 4 − 3i,then limn→∞zn · wn = 25. (You may use the fact that convergent complex sequences are bounded.)
- Given that a mountain path is a sequence steps that never go under the x- axis (no negative points) and are sequences of NE(going up one unit and right one unit) and SE(going down one unit and right one unit). Prove that the mountain paths from point (0, 0) -> (2x, 0) are Catalan object. Note: A Catalan object is proven by one of the following (i) a bijection of a known Catalan object (ii) show it satifies a Catalan recursion (iii) directly count it and find there are (1/(x+1))*(2x choose x) of them Each mountain path for x = 3 are shown in attached photo:Give a combinatorial interpretation of the coefficientof x4 in the expansion (1 + x + x2 + x3 +⋯)3. Use thisinterpretation to find this number.1. Consider the recursion ut = 3ut−1 − b. (a) Find the value of b such that the equilibrium solution of this recursion is uˆ = 500. (b) Using the value of b you found in question (a), make a change of variables and show how this change transforms the affine geometric recursion ut = 3ut−1 − b to a geometric recursion of the form vt = λvt−1.