# 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 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 $6$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 $13$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 $6 n - 1$

- 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

*Previous ... Next*: Sequence of Fibonacci Numbers ending in Index*Previous ... Next*: Sequence of Powers of 5

*Previous ... Next*: Numbers not Sum of Square and Prime*Previous ... Next*: Smallest Arguments for given Multiplicative Persistence

*Previous ... Next*: Powerful Number*Previous ... Next*: Square Number*Previous ... Next*: Squares with No More than 2 Distinct Digits

*Previous ... Next*: Lucky Number

*Previous ... Next*: Semiprime Number

*Previous ... Next*: 91 is Pseudoprime to 35 Bases less than 91*Previous ... Next*: Odd Numbers Not Expressible as Sum of 4 Distinct Non-Zero Coprime Squares

*Previous ... Next*: Integers not Expressible as Sum of Distinct Primes of form 6n-1*Previous ... Next*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers*Previous ... Next*: Positive Integers Not Expressible as Sum of Distinct Non-Pythagorean Primes*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation*Previous ... Next*: Trimorphic Number

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $25$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2025$ - 1992: David Wells:
*Curious and Interesting Puzzles*... (previous) ... (next): Egyptian Fractions: $5$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $25$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2025$

Categories:

- Powers of 5/Examples
- Automorphic Numbers/Examples
- Cullen Numbers/Examples
- Multiplicative Persistence/Examples
- Powerful Numbers/Examples
- Square Numbers/Examples
- Lucky Numbers/Examples
- Semiprimes/Examples
- Integers not Expressible as Sum of Distinct Primes of form 6n-1/Examples
- Trimorphic Numbers/Examples
- Specific Numbers
- 25