# 25

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## Number

$25$ (twenty-five) is:

$5^2$

The only integer satisfying the equation $\paren {n - 1}! + 1 = n^k$:
$25 = 4! + 1 = 5^2$

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

The $1$st of the only known pair of consecutive odd powerful numbers, the other being $27$:
$25 = 5^2$, $27 = 3^3$

The $1$st square number which is the sum of two square numbers:
$25 = 16 + 9 = 4^2 + 3^2 = 5^2$

The $1$st positive integer having a multiplicative persistence of $2$.

The $2$nd power of $5$ after $(1)$, $5$:
$25 = 5^2$

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

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

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

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

The $4$th non-negative integer $n$ after $0$, $1$, $5$ such that the Fibonacci number $F_n$ ends in $n$

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

The $5$th square after $1$, $4$, $9$, $16$ which has no more than $2$ distinct digits

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

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

The $10$th of $35$ integers less than $91$ to which $91$ itself is a Fermat pseudoprime:
$3$, $4$, $9$, $10$, $12$, $16$, $17$, $22$, $23$, $25$, $\ldots$

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 $16$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.

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 number of primes with no more than $2$ digits:
$2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$, $43$, $47$, $53$, $59$, $61$, $67$, $71$, $73$, $79$, $83$, $89$, $97$

Can be expressed as the sum of $n$ non-zero squares for all $n$ from $4$ to $11$.

Adding $1$ to each of its digits yields another square:
$25 + 11 = 36 = 6^2$
The roots of those squares also differ by a repunit:
$5 + 1 = 6$