Number Theory 8.29) Use descent to prove that x3 + 2y3 (0,0,0) 423 has no integer...
I need help with number 3 on my number theory
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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 +...
For Python: A prime number is defined as an integer greater than 1 that has no positive divisors other than 1 and itself. Write a program that prompts the user for an integer > 1. Validate the value is > 1 (if not, ask for another). Use a loop to determine if the number is prime or not. Issue an appropriate message. [complete this part before proceeding]. Add a loop that continues to ask the user if they would like...
Tems.] Use the second principle of induction to prove that every positive integer n has a factorization of the form 2m, where m is odd. (Hint: For n > 1, n is either odd or is divisible by 2.)
A firm has prepared the following binary integer program to evaluate a number of potential locations for new warehouses. The firm's goal is to maximize the net present value of their decision while not spending more than their currently available capital. Max 20x1 + 30x2 + 10x3 + 15x4 s.t. 5x1 + 7x2 + 12x3 + 11x4 ≤ 21 {Constraint 1} x1 + x2 + x3 + x4 ≥ 2 {Constraint 2} x1 + x2 ≤ 1 {Constraint 3} x1...
Prove using mathematical induction that for every positive integer n, = 1/i(i+1) = n/n+1. 2) Suppose r is a real number other than 1. Prove using mathematical induction that for every nonnegative integer n, = 1-r^n+1/1-r. 3) Prove using mathematical induction that for every nonnegative integer n, 1 + i+i! = (n+1)!. 4) Prove using mathematical induction that for every integer n>4, n!>2^n. 5) Prove using mathematical induction that for every positive integer n, 7 + 5 + 3 +.......
10. Use 9 above to prove that the equation x^2 − 2y^2 = 1 has infinitely many solutions over Q. What can you conclude about the number of solutions over Z? (question9: For F as in 8, define N : F → Q by N(a + b√2) = a^2 − 2b^2. (i) Prove that N(αβ) = N(α)N(β), for all α,β ∈ F. (ii) Find an element u ∈ F such that N(u) = 1 and such that all of the...
Use simulations to prove that the binomial distribution is correct. The binomial distribution has two parameters n and p. There are n trials and each has two possible outcomes, with probability p for “success” and 1-p for “failure”. The binomial gives the probability distribution for the number of successes in n trials. You will conduct simulations with r replicates, where each simulation replicates does n simulated “coin flips”. You will add up the number of successes in each coin flip,...
Question #9 all parts thanks
9. The wavefunction, p(x,t), of a particle moving along the x-axis, whose potential energy V(x) is independent of time, is described by the one-dimensional non-relativistic Schrödinger equation (where m is its mass, h is the reduced Planck constant, i is the imaginary number): 2m (a) Verify that it is a parabolic equation (page E-1-2). [It has wave-like solutions, however.] (b) Use the substitution Px,t)-Xx)Tt) to separate the equation into two ODEs. (c) Solve for T,...
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Problem 3: Use simple kinetic theory of gases discussed in section 1.3.2 as well as Fourer's law of condustion to prove: 2 R373 D11 = 3113/202pm Dal We were unable to transcribe this imageof a nes. the xed the led negligible The following assumptions about the structure of the cases are made in order to investigate the statistical rules of the random motion of the molecules: The size of the gas molecules is negligible compared with the distance...
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i pmu r the number of objøcts or ways. Leave your answers in fornsiala form, such as C(3, 2) nporkan?(2) Are repeats poasib Two points each imal digits will have at least one xpeated digin? I. This is the oounting problem Al ancmher so ask yourelr (1) ls onder ipo n How many strings of four bexadeci ) A Compuir Science indtructor has a stack of blue can this i For parts c, d. and e, suppose...