25

Number
$25$ (twenty-five) is:


 * $5^2$


 * The $5$th square number after $1, 4, 9, 16$:
 * $25 = 5 \times 5$


 * The smallest square number which is the sum of two square numbers:
 * $25 = 16 + 9 = 4^2 + 3^2 = 5^2$


 * The $9$th semiprime after $4, 6, 9, 10, 14, 15, 21, 22$:
 * $25 = 5 \times 5$


 * The $4$th automorphic number after $1, 5, 6$:
 * $25^2 = 6 \mathbf {25}$


 * The $7$th trimorphic number after $1, 4, 5, 6, 9, 24$:
 * $25^3 = 15 \, 6 \mathbf {25}$


 * The $8$th lucky number:
 * $1, 3, 7, 9, 13, 15, 21, 25, \ldots$


 * The $3$rd square lucky number:
 * $1, 9, 25, \ldots$


 * The $4$th Cullen number after $1, 3, 9$:
 * $25 = 3 \times 2^3 + 1$


 * The only integer satisfying the equation $\left({n - 1}\right)! + 1 = n^k$:
 * $25 = 4! + 1 = 5^2$


 * The only square number which is $2$ less than a cube:
 * $25 = 3^3 - 2$


 * The $6$th powerful number after $1, 4, 8, 9, 16$


 * The first of the only known pair of consecutive odd powerful numbers, the other being $27$:
 * $25 = 5^2, 27 = 3^3$


 * The $15$th after $1, 2, 4, 5, 6, 8, 9, 12, 13, 15, 16, 17, 20, 24$ of the $24$ positive integers which cannot be expressed as the sum of distinct non-pythagorean primes.


 * The $18$th integer $n$ after $0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15, 16, 18, 19, 24$ such that $2^n$ contains no zero in its decimal representation:
 * $2^{25} = 33 \, 554 \, 432$


 * The $3$rd positive integer which cannot be expressed as the sum of a square and a prime:
 * $1, 10, 25, \ldots$


 * The $16$th positive integer after $2, 3, 4, 7, 8, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24$ which cannot be expressed as the sum of distinct pentagonal numbers.

Also see

 * Consecutive Odd Powerful Numbers
 * Square which is 2 Less than Cube
 * Power of n equalling (n - 1)! + 1