25

== 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 Friedman number base $10$:
::$25 = 5^2$
: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 smallest $n$ such that the Egyptian fraction expansion of $\dfrac 3 n$ using Fibonacci's Greedy Algorithm produces a sequence of $3$ terms when in fact $2$ are sufficient:
::$\dfrac 3 {25} = \dfrac 1 9 + \dfrac 1 {113} + \dfrac 1 {25, 425}$ whereas $\dfrac 3 {25} = \dfrac 1 {10} + \dfrac 1 {50}$
:The $2$nd power of $5$ after $(1)$, $5$:
::$25 = 5^2$
:The number of primes with no more than $2$ digits:
::$2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$, $37$, $41$, $43$, $47$, $53$, $59$, $61$, $67$, $71$, $73$, $79$, $83$, $89$, $97$
: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 and does not end in $0$:
::$25 = 5^2$
: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 $8$th positive integer after $6$, $9$, $12$, $18$, $19$, $20$, $24$ whose cube can be expressed as the sum of $3$ positive cube numbers:
::$25^3 = 4^3 + 17^3 + 22^3$
: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 $13$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
::$1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $25$, $\ldots$
: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 $19$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $8$, $9$, $10$, $12$, $13$, $14$, $18$, $19$, $20$, $21$, $24$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$
: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:
::$5 + 1 = 6$
== Also see ==
* Consecutive Odd Powerful Numbers
* Square which is 2 Less than Cube
* Power of n equalling (n - 1)! + 1
* 25 as Sum of 4 to 11 Squares
* Difference between Two Squares equal to Repunit
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== Sources ==
* : $25$
* : $2025$
* : Egyptian Fractions: $5$
* : $25$
* : $2025$
Category:Specific Numbers
Category:25