n2(n2−1)(n2−4)n^2(n^2-1)(n^2-4) is always divisible by 360 (n>2,n∈N)(n>2,n\in \mathbb{N})

How does one prove that n2(n2−1)(n2−4) is always divisible by 360? (n>2,n∈N) I explain my own way: You can factorize it and get n2(n−1)(n+1)(n−2)(n+2). Then change the condition (n>2,n∈N) into (n>0,n∈N) that is actually equal to (n∈N). Now the statement changes into : n(n+1)(n+2)2(n+3)(n+4) Then I factorized 360 and got 32⋅23⋅5. I don’t know how … Read more

Prove that every number n≥12n \geq 12 is the sum of two composite numbers [closed]

Closed. This question does not meet Mathematics Stack Exchange guidelines. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for Mathematics Stack Exchange. Closed 3 years ago. Improve this question Prove that any natural number greater than or equal to 12 is the sum of two composite … Read more

Prove that every number n≥12n \geq 12 is the sum of two composite numbers [closed]

Closed. This question does not meet Mathematics Stack Exchange guidelines. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for Mathematics Stack Exchange. Closed 3 years ago. Improve this question Prove that any natural number greater than or equal to 12 is the sum of two composite … Read more

Number Theory Taxicab Number

How to prove that there are infinite taxicab numbers? ok i was reading this http://en.wikipedia.org/wiki/Taxicab_number#Known_taxicab_numbers and thought of this question..any ideas? Answer It is easy to show that there are infinitely many positive integers which are representable as the sum of two cubes, e.g., see the article Characterizing the Sum of Two Cubes by K.A. … Read more

Proving an expression is perfect square

I have this expression I got in one larger exercise: (2+√3)2n+1+(2−√3)2n+1−46(2+√3)2(n+1)+1+(2−√3)2(n+1)+1−46+1 and i need to prove it is perfect square. I tried many different approaches but couldn’t find way to show it is square. Interesting fact is (2+√3)(2−√3)=1 so I tried replacing (2+√3)=x and (2−√3)=1/x to see if I would get an idea. Alternative form … Read more

A problem I didn’t know since high school algebra

Determine all positive integers which can be written as a sum of two squares of integers. This is a problem I saw when I was in high school… sum of two squares of integers can be (4k, 4k+1, 4k+2, but no 4k+3?) Answer The square of an even number is a multiple of $4$ (i.e. … Read more

Subtraction of a number

I have a number $x$. If I remove the last digit, I get $y$. Given $x-y$, how can I find $x$? For example x=34 then y=3 given 34-3=31, I have to find 34. if x=4298 then y=429 , given 4298-429 = 3869 . how can I find 4298 from given 3869? Answer Removing the last … Read more

Remainder of a power tower under modulo $2013$

I have an expression like this: $$\left(\large 6000^{5999^{5998^{5997^{{\ldots^{1}}}}}}\right)\bmod 2013$$ Then which method should I use to solve it? Please provide the method not the answer. Editor’s Note: Note that this is a power tower with different values and not the same value as with general tetration. Also, don’t confuse tetration with exponentiation. Both are completely … Read more

Need to solve x^4≡4(mod19)

I need some help in trying to solve : $$x^4\equiv 4\pmod{19}.$$ I have the solution $6$ and $13$ but I’m not clear how it was solved. Answer Since $\mathbb{F}_{19}$ is a field the solutions of $$x^4\equiv 4\pmod{19}$$ are simply given by the solutions of $x^2\equiv 2\pmod{19}$ plus the solutions of $x^2\equiv -2\pmod{19}$. Since … Read more

If x^2\equiv a\pmod nx^2\equiv a\pmod n, then (n-x)^2\equiv a\pmod n(n-x)^2\equiv a\pmod n

Given that x is a solution to x^{2}\equiv a \pmod n, show that y=n-x is also a solution. Please don’t solve, just give me a hint. Answer Hint Expand y^2 = (n – x)^2 modulo n. (This is perhaps easier to see intuitively if we recall that n – x \equiv -x \bmod n, in … Read more