Let 1 1+ + + + 2 3 1 r 823 823s Without calculating the left-hand side, Find one solution of the polynomial congruence 3x²+2x+100 = 0 (mod 343). Ts (mod 8233).

By considering appropriate series expansions, prove that ez · e²²/2 . e²³/3 . ... = 1 + x + x² + · ·. when <1.

Prove that Σ prime p≤x p=3 (mod 10) 1 Р = for some constant A. log log x + A+O 1 log x ,

Let Σ 1 and g(x) = Σ logp. f(x) = prime p≤x p=3 (mod 10) prime p≤x p=3 (mod 10) g(x) = f(x) logx - Ր _☑ t¯¹ƒ(t) dt. Assuming that f(x) ~ 1½π(x), prove that g(x) ~ 1x. 米 (You may assume the Prime Number Theorem: 7(x) ~ x/log x.) *

Let Σ logp. f(x) = Σ 1 and g(x) = Σ prime p≤x p=3 (mod 10) (i) Find ƒ(40) and g(40). prime p≤x p=3 (mod 10) (ii) Prove that g(x) = f(x) logx – [*t^¹ƒ(t) dt. 2

When P is True and Q is False, what is the truth value of (P→ ~Q)? a. True ○ b. False c. unknown

No chatgpt pls

you stop recording velocity data at t = 4.0s but you notice a short time later that your friend going fast at 13.8m/s. assuming he keeps the same acceleration, how much time passed since you stopped recording data

Given sets X and Y with X ∈ Y, is it always true that P (X) ∈ P (Y) (power sets)? If not, what is  a counterexample?