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Numbers $n$ such that the digit sum of $n^2$ is a square

number-theory decimal-expansion square-numbers asked Jan 6 '15 at 17:33

math.stackexchange.com

69 votes

Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

number-theory rational-numbers perfect-powers asked Mar 11 '15 at 16:44

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54 votes

Finding triplets $(a,b,c)$ such that $\sqrt{abc}\in\mathbb N$ divides $(a-1)(b-1)(c-1)$

elementary-number-theory diophantine-equations radicals asked Apr 25 '15 at 17:48

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53 votes

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

number-theory graph-theory exponentiation perfect-powers asked Feb 28 '15 at 12:04

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47 votes

Why does this ratio of sums of square roots equal $1+\sqrt2+\sqrt{4+2\sqrt2}=\cot\frac\pi{16}$ for any natural number $n$?

sequences-and-series geometry trigonometry summation fractions asked Aug 10 '13 at 15:07

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42 votes

$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true?

number-theory functions summation radicals floor-function asked Aug 27 '13 at 7:05

math.stackexchange.com

40 votes

Numbers $n$ such that the digit sums of $n, n^2,\cdots,n^k$ coincide.

number-theory decimal-expansion asked Jan 14 '15 at 12:36

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38 votes

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

inequality exponential-function exponentiation pi number-comparison asked Aug 26 '15 at 12:10

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37 votes

Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$

analysis trigonometry summation closed-form fractions asked Oct 29 '13 at 14:40

math.stackexchange.com

33 votes

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

mg.metric-geometry euclidean-geometry plane-geometry polyhedra asked Oct 1 '13 at 8:29

mathoverflow.net

Top Answers

131

Mental Calculations

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How to prove that the Fibonacci sequence is periodic mod 5 without using induction?

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Trick with 3-digit numbers, always get 1089

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Which of the following powers is bigger: $2^{41}$ or $3^{24}$?

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Are Mersenne prime exponents always odd?

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Simplify $\left({\sum_{k=1}^{2499}\sqrt{10+{\sqrt{50+\sqrt{k}}}}}\right)\left({\sum_{k=1}^{2499}\sqrt{10-{\sqrt{50+\sqrt{k}}}}}\right)^{-1}$

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Proving the following number is real

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Compare two powers of numbers without common divisor

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Which number is bigger: $\sqrt[10]{2}$ or $1.2$?

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Can we simplify $\int_{0}^{\infty}\frac{{\sin}^px}{x^q}dx$?

mathoverflow.net

Is there a closed form for the polynomials satisfying this recurrence relation?

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Prove by contradiction that $\forall x,y \in \Bbb Z: x^2-4y \ne 2$

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$\int_{0}^{\infty}{\left(x^3\cdot(e^x-1)^{-1} \right)}dx$

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Make $20$ using two “$3$”s

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Is there another way to solve this quadratic equation?

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What is the ten's digit of $7^{7^{7^{7^7}}}$

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What is the value of $\frac{a^3}{a^6+a^5+a^4+a^3+a^2+a+1}$

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Evaluate $a^2+b^2+c^2$

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Finding $\binom{999}{0}-\binom{999}{2}+\binom{999}{4}-\binom{999}{6}+\cdots +\binom{999}{996}-\binom{999}{998}$

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Reverse the equation to solve for y

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find the largest number

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Right triangle inscribed in a square. Find the square area?

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Integrating a rational function.

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Deciding whether a number is rational (2 examples)

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Expression with last digits different

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Prove that a logarithm is irrational

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Evaluating the $\lim_{x\to0}\frac{4^x-1}{8^x-1}$ without L'Hopital Rule

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Do there exist four polynomials with real coefficients so that …

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What is the name of the circle that is tangent to three mutually-tangent circles centered at the vertices of a triangle?

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Solving a special Quartic Equation.

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Numbers $n$ such that the digit sum of $n^2$ is a square

Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

Finding triplets $(a,b,c)$ such that $\sqrt{abc}\in\mathbb N$ divides $(a-1)(b-1)(c-1)$

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

Why does this ratio of sums of square roots equal $1+\sqrt2+\sqrt{4+2\sqrt2}=\cot\frac\pi{16}$ for any natural number $n$?

$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true?

Numbers $n$ such that the digit sums of $n, n^2,\cdots,n^k$ coincide.

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals