Found problems: 521
Find all positive integers $n$ such that $27^n- 2^n$ is a perfect square.
Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.
Find the number of integers $n$ between $1$ and $2021$ such that $2^n+2^{n+3}$ is a perfect square.
A positive integer $N$ has the digits $1, 2, 3, 4, 5, 6$ and $7$, so that each digit $i$, $i \in \{1, 2, 3, 4, 5, 6, 7\}$ occurs $4i$ times in the decimal representation of $N$. Prove that $N$ is not a perfect square.
Find the sum of all non-zero digits that can repeat at the end of a perfect square. (For example, if $811$ were a perfect square, $1$ would be one of these non-zero digits.)
Prove that the product of five consecutive positive integers cannot be the square of an integer.
Consider the sequence $a(n)$ defined by the following conditions:$$a(1) = 1\,\,\,\, a(n + 1) = a(n) + [\sqrt{a(n)}] \,\,\, , \,\,\,\, n = 1,2,3,...$$ How many perfect squares no greater in value than $1000 000$ will be found among the first terms of the sequence? ( (Note: $[x]$ means the integer part of $x$, that is the greatest integer not greater than $x$.)
(A Andjans)
Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000 \cdot N$ contains no square of an integer.
Let $a^2_1+ a^2_2+ a^2_3+ a^2_4+ a^2_5= b^2$, where $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, and $b$ are integers. Prove that not all of these numbers can be odd.
Find all positive integers $n$ such that $n^3 - 18n^2 + 115n - 391$ is the cube of a positive intege
The sequence $x_{n}$ is de fined by:
$x_{0}=1, x_{1}=0, x_{2}=1,x_{3}=1, x_{n+3}=\frac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\frac{n+1}{n}x_{n} (n=1,2,3..)$
Prove that all members of the sequence are perfect squares.
(a) Let $a$ and $b$ be positive integers such that $M(a, b) = a - \frac1b +b(b + \frac3a)$ is an integer.
Prove that $M(a,b)$ is a square.
(b) Find nonzero integers $a$ and $b$ such that $M(a,b)$ is a positive integer, but not a square.
Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$.
Prove that for no $n$: $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square.
Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$. Prove that $S_1 \cdot S_2$ is divisible by $150$.
Determine all primes $p$ such that $5^p + 4 p^4$ is a perfect square, i.e., the square of an integer.
Let $n$ be an even positive integer. We fill in a number on each cell of a rectangle table of $n$ columns and multiple rows as following:
i. Each row is assigned to some positive integer $a$ and its cells are filled by $0$ or $a$ (in any order);
ii. The sum of all numbers in each row is $n$.
Note that we cannot add any more row to the table such that the conditions (i) and (ii) still hold.
Prove that if the number of $0$’s on the table is odd then the maximum odd number on the table is a perfect square.
Let $n>2$ be a positive integer and $b=2^{2^n}$. Let $a$ be an odd positive integer such that $a\le b \le 2a$.
Show that $a^2+b^2-ab$ is not a square.
Find the largest natural number $n$ for which $4^{995} +4^{1500} +4^n$ is a square.
Find three different positive integers whose sum is minimum than meet the condition that the sum of each pair of them is a perfect square.
Find all natural numbers $n$ for which $2^8 +2^{11} +2^n$ is a perfect square.
Find all integer values of $x$ for which the value of the expression
\[x^2+6x+33\]
is a perfect square.
Determine all positive integers $n < 200$, such that $n^2 + (n+ 1)^2$ is the square of an integer.
Sasha wrote a non-zero number on the board and added it to it on the right, one non-zero digit at a time, until he writes out a million digits. Prove that an exact square has been written on the board no more than $100$ times.
Find all integers$ n$ such that $n^2 + 24n + 35$ is a square.
Determine all naturals $n$ such that $2^n + 5$ is a perfect square.
a) Exist $a, b, c, \in N$, such that the numbers $ab+1,bc+1$ and $ca+1$ are simultaneously even perfect squares ?
b) Show that there is an infinity of natural numbers (distinct two by two) $a, b, c$ and $d$, so that the numbers $ab+1,bc+1, cd+1$ and $da+1$ are simultaneously perfect squares.