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 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 $6 n - 1$
- 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
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- Previous ... Next: Square Number
- Previous ... Next: Squares with No More than 2 Distinct Digits
- Previous ... Next: Lucky Number
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- 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: Cubes which are Sum of Three Cubes
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
- Friedman Numbers Base 10/Examples
- Specific Numbers
- 25