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# 25

Please do not rely on any information it contains.

25 is an integer, the smallest square that can be written as a sum of 2 (nonzero) squares (A134422). It is also the smallest composite number that is both a lucky number (A000959) and ludic number (A003309). Also, the smallest composite number, whose binary expansion is 11001, such that when interpreted as a polynomial in ring ${\displaystyle GF(2)[X],(x^{4}+x^{3}+1)}$, is irreducible in that ring (A091214).

## Membership in core sequences

 Odd numbers ..., 19, 21, 23, 25, 27, 29, 31, ... A005408 Semiprimes ..., 15, 21, 22, 25, 26, 33, 34, ... A001358 Perfect squares ..., 4, 9, 16, 25, 36, 49, 64, 81, ... A000290 Powers of primes ..., 17, 19, 23, 25, 27, 29, 31, ... A000961 Composite numbers ..., 21, 22, 24, 25, 26, 27, 28, ... A002808 Lucky numbers ..., 13, 15, 21, 25, 31, 33, 37, ... A000959

In Pascal's triangle, 25 occurs twice.

## Sequences pertaining to 25

 Multiples of 25 0, 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, ... A008607 ${\displaystyle 3x+1}$ sequence beginning at 33 33, 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, ... A008880 ${\displaystyle 3x-1}$ sequence beginning at 17 17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61, 182, 91, ... A003124

## Partitions of 25

There are 1958 partitions of 25.

The Goldbach represenations of 25 using distinct primes are: 2 + 23 = 3 + 5 + 17 = 5 + 7 + 13 = 25.

## Roots and powers of 25

In the table below, irrational numbers are given truncated to eight decimal places.

 ${\displaystyle {\sqrt {25}}}$ 5.00000000 25 2 625 ${\displaystyle {\sqrt[{3}]{25}}}$ 2.92401773 A010597 25 3 15625 ${\displaystyle {\sqrt[{4}]{25}}}$ 2.23606797 A002163 25 4 390625 ${\displaystyle {\sqrt[{5}]{25}}}$ 1.90365393 A011110 25 5 9765625 ${\displaystyle {\sqrt[{6}]{25}}}$ 1.70997594 A005481 25 6 244140625 ${\displaystyle {\sqrt[{7}]{25}}}$ 1.58381960 25 7 6103515625 ${\displaystyle {\sqrt[{8}]{25}}}$ 1.49534878 A011003 25 8 152587890625 ${\displaystyle {\sqrt[{9}]{25}}}$ 1.42996914 25 9 3814697265625 ${\displaystyle {\sqrt[{10}]{25}}}$ 1.37972966 A005534 25 5 95367431640625 A009969

## Logarithms and 25th powers

In the OEIS specifically and mathematics in general, ${\displaystyle \log x}$ refers to the natural logarithm of ${\displaystyle x}$, whereas all other bases are specified with a subscript.

As above, irrational numbers in the following table are truncated to eight decimal places.

 ${\displaystyle \log _{25}2}$ 0.21533827 ${\displaystyle \log _{2}25}$ 4.64385618 2 25 33554432 ${\displaystyle \log _{25}e}$ 0.62133493 ${\displaystyle \log 25}$ 1.60943791 A016628 ${\displaystyle e^{25}}$ 72004899337.38587252 ${\displaystyle \log _{25}3}$ 0.34130309 ${\displaystyle \log _{3}25}$ 2.92994704 A228375 3 25 847288609443 ${\displaystyle \log _{25}\pi }$ ${\displaystyle \log _{\pi }25}$ ${\displaystyle \pi ^{25}}$ ${\displaystyle \log _{25}4}$ 0.43067655 A152675 ${\displaystyle \log _{4}25}$ 2.32192809 A020858 4 25 1125899906842624 ${\displaystyle \log _{25}5}$ 0.50000000 A020761 ${\displaystyle \log _{5}25}$ 2.00000000 A000038 5 25 298023223876953125 ${\displaystyle \log _{25}6}$ 0.55664137 ${\displaystyle \log _{6}25}$ 1.79648880 6 25 28430288029929701376 ${\displaystyle \log _{25}7}$ 0.60453097 ${\displaystyle \log _{7}25}$ 1.65417495 7 25 1341068619663964900807 ${\displaystyle \log _{25}8}$ 0.64601483 ${\displaystyle \log _{8}25}$ 1.54795206 8 25 37778931862957161709568 ${\displaystyle \log _{25}9}$ 0.68260619 A152914 ${\displaystyle \log _{9}25}$ 1.46497352 A113209 9 25 717897987691852588770249 ${\displaystyle \log _{25}10}$ 0.71533827 ${\displaystyle \log _{10}25}$ 1.39794000 10 25 10000000000000000000000000

(See A010813 for the 25th powers of integers).

## Values for number theoretic functions with 25 as an argument

 ${\displaystyle \mu (25)}$ 0 ${\displaystyle M(25)}$ –2 ${\displaystyle \pi (25)}$ 9 ${\displaystyle \sigma _{1}(25)}$ 31 ${\displaystyle \sigma _{0}(25)}$ 3 ${\displaystyle \phi (25)}$ 20 ${\displaystyle \Omega (25)}$ 2 ${\displaystyle \omega (25)}$ 1 ${\displaystyle \lambda (25)}$ 20 This is the Carmichael lambda function. ${\displaystyle \lambda (25)}$ 1 This is the Liouville lambda function. ${\displaystyle \zeta (25)}$ 25! 15511210043330985984000000 ${\displaystyle \Gamma (25)}$ 620448401733239439360000

## Factorization of some small integers in a quadratic integer ring adjoining the cubic root of 5

Since 25 is a perfect square, adjoining its square root to the field of rational numbers does not generate a field extension.

However, adjoining the cubic root of 5 to ${\displaystyle \mathbb {Q} }$ does generate a field extension, one in which the algebraic integers are of the form ${\displaystyle a+b{\sqrt[{3}]{5}}+c{\sqrt[{3}]{25}}}$, with ${\displaystyle a,b,c}$ being integers from ${\displaystyle \mathbb {Z} }$. The norm function is then ${\displaystyle a^{3}+5b^{3}+25c^{3}-15abc}$.

These algebraic integers form a ring that is a unique factorization domain. The following table shows a few small primes factorized in this ring.

TABLE GOES HERE

## Factorization of 25 in some quadratic integer rings

In ${\displaystyle \mathbb {Z} }$, 25 is the square of 5. But it has different factorizations in some quadratic integer rings, and in a few cases it's not a simple matter of taking the factorization of 5 and adding in some exponent 2s.

 ${\displaystyle \mathbb {Z} [i]}$ ${\displaystyle (2-i)^{2}(2+i)^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-2}}]}$ 5 2 ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$ 5 2 ${\displaystyle \mathbb {Z} [\omega ]}$ ${\displaystyle \mathbb {Z} [{\sqrt {3}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$ ${\displaystyle ({\sqrt {-5}})^{4}}$ ${\displaystyle \mathbb {Z} [\phi ]}$ ${\displaystyle (-1+2\phi )^{4}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-6}}]}$ 5 2 OR ${\displaystyle (1-2{\sqrt {-6}})(1+2{\sqrt {-6}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {6}}]}$ ${\displaystyle (1-{\sqrt {6}})^{2}(1+{\sqrt {6}})^{2}}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-7}})}}$ 5 2 ${\displaystyle \mathbb {Z} [{\sqrt {7}}]}$ 5 2 ${\displaystyle \mathbb {Z} [{\sqrt {-10}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {10}}]}$ 5 2 OR ${\displaystyle (-1)(15-5{\sqrt {10}})(15+5{\sqrt {10}})}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-11}})}}$ ${\displaystyle \left({\frac {3}{2}}-{\frac {\sqrt {-11}}{2}}\right)^{2}\left({\frac {3}{2}}+{\frac {\sqrt {-11}}{2}}\right)^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {11}}]}$ ${\displaystyle (4-{\sqrt {11}})^{2}(4+{\sqrt {11}})^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {-13}}]}$ 5 2 ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {13}})}}$ 5 2 ${\displaystyle \mathbb {Z} [{\sqrt {-14}}]}$ ${\displaystyle \mathbb {Z} [{\sqrt {14}}]}$ ${\displaystyle (3-{\sqrt {14}})^{2}(3+{\sqrt {14}})^{2}}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-15}})}}$ ${\displaystyle \mathbb {Z} [{\sqrt {15}}]}$ 5 2 OR ${\displaystyle (20-5{\sqrt {15}})(20+5{\sqrt {15}})}$ ${\displaystyle \mathbb {Z} [{\sqrt {-17}}]}$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {17}})}}$ 5 2 ${\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {-19}})}}$ ${\displaystyle \left({\frac {1}{2}}-{\frac {\sqrt {-19}}{2}}\right)^{2}\left({\frac {1}{2}}+{\frac {\sqrt {-19}}{2}}\right)^{2}}$ ${\displaystyle \mathbb {Z} [{\sqrt {19}}]}$ ${\displaystyle (9-2{\sqrt {19}})^{2}(9+2{\sqrt {19}})^{2}}$.

Note that −1 does not appear in the factorization of 25 in ${\displaystyle \mathbb {Z} [{\sqrt {-5}}]}$, Since ${\displaystyle ({\sqrt {-5}})^{2}=-5}$ and ${\displaystyle (-5)^{2}=25}$.

## Representation of 25 in various bases

 Base 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Representation 11001 221 121 100 41 34 31 27 25 23 21 1C 1B 1A 19 18 17 16 15

 ${\displaystyle -1}$ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1729