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Tutor profile: John C.

John C.
Problem Solver

Questions

Subject: Number Theory

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Question:

The prime twins $$5$$ and $$7$$ have a triangular number between them. Are there any other triangles $$t$$ such that $$t-1$$ and $$t+1$$ are both prime?

John C.
Answer:

No. A triangle takes the form $$t=\dfrac {n(n+1)}2$$. Subtracting $$1$$ gives $$\dfrac {(n-1)(n+2)}2$$. One of $$n-1$$ and $$n+2$$ must be even, so $$\dfrac {(n-1)(n+2)}2$$ is the product of two positive integers if $$n>1$$. It can only be prime if $$\dfrac {n-1}2=1$$, in which case $$t=6$$, or $$n-1=1$$, in which case $$t=3$$.

Subject: Calculus

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Question:

Prove that $$e^x\ge x+1$$ for all real $$x$$.

John C.
Answer:

Consider the function $$f(x) = e^x-x-1$$, and find its minimum by the second derivative test. The first derivative is $$\displaystyle \frac {\mathrm df}{\mathrm dx}=e^x-1$$ and the second derivative is $$\displaystyle \frac {\mathrm d^2f}{\mathrm dx^2}=e^x$$ If there are any critical points, they will come where the first derivative is 0 - that is, when $$e^x=1$$. But then the second derivative is positive, so this is a minimum. Since $$e^x=1$$ implies $$x = \ln 1= 0$$, we have $$f(x) \ge f(0)=0$$ for all real $$x$$, or$$e^x\ge x+1$$.

Subject: Algebra

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Question:

Let $$r$$ and $$s$$ be the roots of the quadratic equation $$ax^2+bx+c=0$$. If $$c \ne 0$$, what are the roots of $$cy^2+by+a=0$$?

John C.
Answer:

Since $$c \ne 0$$, $$r$$ and $$s$$ are not equal to $$0$$, so we can divide the original equation by $$x^2$$, giving $$a+b \frac 1x +c \frac 1{x^2}=0$$. Substituting $$y=1/x$$, we see that the roots of the second equation are $$1/r$$ and $$1/s$$.

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