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Q: Generalisation of IMO 1990/P3:For which $b $ do there exist infinitely many positive integers $n$ such that $n^2$ divides $b^n+1$?

Sunaina Pati For which positive integers $b > 2$ do there exist infinitely many positive integers $n$ such that $n^2$ divides $b^n+1$? It was from my LTE/ Zsigmondy handout. By taking examples, it looks like for $b= 2^k-1 , 2$ it's not true . Here's my progress: I got $b=4,5,6,8,9$ works ( $2,3,7$ doesn't ...

 
 
6 hours later…
1:38 PM
In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if a > b > 0 are coprime integers, then for any integer n ≥ 1, there is a prime number p (called a primitive prime divisor) that divides an − bn and does not divide ak − bk for any positive integer k < n, with the following exceptions: n = 1, a − b = 1; then an − bn = 1 which has no prime divisors n = 2, a + b a power of two; then any odd prime factors of a2 - b2 = (a + b)(a1 - b1) must be contained in a1 - b1, which is also even n = 6, a = 2, b = 1; then a6 − b6 = 63 = 32×7 = (a2 − b2)2(a3 − b3)This generalizes Bang...
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Q: Using Excel to keep track of maintenance services done on machines

MichałMAssume that you buy a caterpillar, and for every 10 hours of work, you need to apply lubricant. For every 100, you need to replace a certain part. For 500 hours, you need to perform any given adjustment. For 1000 hours, you need to have it checked by a technician. The formula/s I need should take...

 
 
9 hours later…
10:11 PM
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Q: Solving Hodgkin and Huxley equation

cccI'm trying to solve Hodgkin and Huxley equation given by, $$\frac{d n}{d t}=\alpha_{n}(1-n)-\beta_{n} n$$ using the boundary condition $$n_{0}=\frac{\alpha_{n_{0}}}{\alpha_{n_{0}}+\beta_{n_{0}}}$$. The solution of this differential equation should be $$\begin{aligned} &n=n_{\infty}-\left(n_{\inft...

The Hodgkin–Huxley model, or conductance-based model, is a mathematical model that describes how action potentials in neurons are initiated and propagated. It is a set of nonlinear differential equations that approximates the electrical characteristics of excitable cells such as neurons and cardiac myocytes. It is a continuous-time dynamical system. Alan Hodgkin and Andrew Huxley described the model in 1952 to explain the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon. They received the 1963 Nobel Prize in Physiology or Medicine for this...
 

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