Cube (algebra)

Number raised to the third power

"Third power" redirects here. For other uses, see Third power (disambiguation).

"Cubed" redirects here. For other uses, see Cube (disambiguation).

y = x3 for values of 1 ≤ x ≤ 25.

In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together. The cube of a number n is denoted n3, using a superscript 3,[a] for example 23 = 8. The cube operation can also be defined for any other mathematical expression, for example (x \+ 1)3.

The cube is also the number multiplied by its square:

n3 = n × n2 = n × n × n.

The cube function is the function x ↦ x3 (often denoted y = x3) that maps a number to its cube. It is an odd function, as

(−n)3 = −(n3).

The volume of a geometric cube is the cube of its side length, giving rise to the name. The inverse operation that consists of finding a number whose cube is n is called extracting the cube root of n. It determines the side of the cube of a given volume. It is also n raised to the one-third power.

The graph of the cube function is known as the cubic parabola. Because the cube function is an odd function, this curve has a center of symmetry at the origin, but no axis of symmetry.

## In integers

[edit]

See also: Cube-free integer

A cube number, or a perfect cube, or sometimes just a cube, is a number which is the cube of an integer. The non-negative perfect cubes up to 603 are (sequence A000578 in the OEIS):

03 = | 0

|

13 = | 1  | 113 = | 1331  | 213 = | 9261  | 313 = | 29,791  | 413 = | 68,921  | 513 = | 132,651
23 = | 8  | 123 = | 1728  | 223 = | 10,648  | 323 = | 32,768  | 423 = | 74,088  | 523 = | 140,608
33 = | 27  | 133 = | 2197  | 233 = | 12,167  | 333 = | 35,937  | 433 = | 79,507  | 533 = | 148,877
43 = | 64  | 143 = | 2744  | 243 = | 13,824  | 343 = | 39,304  | 443 = | 85,184  | 543 = | 157,464
53 = | 125  | 153 = | 3375  | 253 = | 15,625  | 353 = | 42,875  | 453 = | 91,125  | 553 = | 166,375
63 = | 216  | 163 = | 4096  | 263 = | 17,576  | 363 = | 46,656  | 463 = | 97,336  | 563 = | 175,616
73 = | 343  | 173 = | 4913  | 273 = | 19,683  | 373 = | 50,653  | 473 = | 103,823  | 573 = | 185,193
83 = | 512  | 183 = | 5832  | 283 = | 21,952  | 383 = | 54,872  | 483 = | 110,592  | 583 = | 195,112
93 = | 729  | 193 = | 6859  | 293 = | 24,389  | 393 = | 59,319  | 493 = | 117,649  | 593 = | 205,379
103 = | 1000  | 203 = | 8000  | 303 = | 27,000  | 403 = | 64,000  | 503 = | 125,000  | 603 = | 216,000

Geometrically speaking, a positive integer m is a perfect cube if and only if one can arrange m solid unit cubes into a larger, solid cube. For example, 27 small cubes can be arranged into one larger one with the appearance of a Rubik's Cube, since 3 × 3 × 3 = 27.

The difference between the cubes of consecutive integers can be expressed as follows:

n3 − (n − 1)3 = 3(n − 1)n \+ 1.

(n \+ 1)3 − n3 = 3(n \+ 1)n \+ 1.

There is no minimum perfect cube, since the cube of a negative integer is negative. For example, (−4) × (−4) × (−4) = −64.

### Base ten

[edit]

Unlike perfect squares, perfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 25, 75 and 00 can be the last two digits, any pair of digits with the last digit odd can occur as the last digits of a perfect cube. With even cubes, there is considerable restriction, for only 00, o2, e4, o6 and e8 can be the last two digits of a perfect cube (where o stands for any odd digit and e for any even digit). Some cube numbers are also square numbers; for example, 64 is a square number (8 × 8) and a cube number (4 × 4 × 4). This happens if and only if the number is a perfect sixth power (in this case 26).

The last digits of each 3rd power are:

0 | 1 | 8 | 7 | 4 | 5 | 6 | 3 | 2 | 9 |

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It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have digital root 1, 8 or 9. That is their values modulo 9 may be only 0, 1, and 8. Moreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3:

* If the number x is divisible by 3, its cube has digital root 9; that is,

if x ≡ 0 ( mod 3 ) then x 3 ≡ 0 ( mod 9 ) (actually 0 ( mod 27 ) ) ; {\displaystyle {\text{if}}\quad x\equiv 0{\pmod {3}}\quad {\text{then}}\quad x^{3}\equiv 0{\pmod {9}}{\text{ (actually}}\quad 0{\pmod {27}}{\text{)}};}
* If it has a remainder of 1 when divided by 3, its cube has digital root 1; that is,

if x ≡ 1 ( mod 3 ) then x 3 ≡ 1 ( mod 9 ) ; {\displaystyle {\text{if}}\quad x\equiv 1{\pmod {3}}\quad {\text{then}}\quad x^{3}\equiv 1{\pmod {9}};}
* If it has a remainder of 2 when divided by 3, its cube has digital root 8; that is,

if x ≡ 2 ( mod 3 ) then x 3 ≡ 8 ( mod 9 ) . {\displaystyle {\text{if}}\quad x\equiv 2{\pmod {3}}\quad {\text{then}}\quad x^{3}\equiv 8{\pmod {9}}.}

### Sums of two cubes

[edit]

Main article: Sum of two cubes

### Sums of three cubes

[edit]

Main article: Sums of three cubes

It is conjectured that every integer (positive or negative) not congruent to ±4 modulo 9 can be written as a sum of three (positive or negative) cubes with infinitely many ways.[1] For example,  6 = 2 3 + ( − 1 ) 3 + ( − 1 ) 3 {\displaystyle 6=2^{3}+(-1)^{3}+(-1)^{3}} . Integers congruent to ±4 modulo 9 are excluded because they cannot be written as the sum of three cubes.

The smallest such integer for which such a sum is not known is 114. In September 2019, the previous smallest such integer with no known 3-cube sum, 42, was found to satisfy this equation:[2]

42 = ( − 80538738812075974 ) 3 + 80435758145817515 3 + 12602123297335631 3 . {\displaystyle 42=(-80538738812075974)^{3}+80435758145817515^{3}+12602123297335631^{3}.}

One solution to  x 3 + y 3 + z 3 = n {\displaystyle x^{3}+y^{3}+z^{3}=n} is given in the table below for n ≤ 78, and n not congruent to 4 or 5 modulo 9. The selected solution is the one that is primitive (gcd(x, y, z) = 1), is not of the form  c 3 + ( − c ) 3 + n 3 = n 3 {\displaystyle c^{3}+(-c)^{3}+n^{3}=n^{3}} or  ( n + 6 n c 3 ) 3 + ( n − 6 n c 3 ) 3 + ( − 6 n c 2 ) 3 = 2 n 3 {\displaystyle (n+6nc^{3})^{3}+(n-6nc^{3})^{3}+(-6nc^{2})^{3}=2n^{3}} (since they are infinite families of solutions), satisfies 0 ≤ |x| ≤ |y| ≤ |z|, and has minimal values for |z| and |y| (tested in this order).[3][4][5]

Only primitive solutions are selected since the non-primitive ones can be trivially deduced from solutions for a smaller value of n. For example, for n = 24, the solution  2 3 + 2 3 + 2 3 = 24 {\displaystyle 2^{3}+2^{3}+2^{3}=24} results from the solution  1 3 + 1 3 + 1 3 = 3 {\displaystyle 1^{3}+1^{3}+1^{3}=3} by multiplying everything by  8 = 2 3 . {\displaystyle 8=2^{3}.} Therefore, this is another solution that is selected. Similarly, for n = 48, the solution (x, y, z) = (−2, −2, 4) is excluded, and this is the solution (x, y, z) = (−23, −26, 31) that is selected.

Primitive solutions for n from 1 to 78

n | x | y | z |  | n | x | y | z
1  | 9 | 10 | −12 | 39  | 117367 | 134476 | −159380
2  | 1214928 | 3480205 | −3528875 | 42  | 12602123297335631 | 80435758145817515 | −80538738812075974
3  | 1 | 1 | 1 | 43  | 2 | 2 | 3
6  | −1 | −1 | 2 | 44  | −5 | −7 | 8
7  | 0 | −1 | 2 | 45  | 2 | −3 | 4
8  | 9 | 15 | −16 | 46  | −2 | 3 | 3
9  | 0 | 1 | 2 | 47  | 6 | 7 | −8
10  | 1 | 1 | 2 | 48  | −23 | −26 | 31
11  | −2 | −2 | 3 | 51  | 602 | 659 | −796
12  | 7 | 10 | −11 | 52  | 23961292454 | 60702901317 | −61922712865
15  | −1 | 2 | 2 | 53  | −1 | 3 | 3
16  | −511 | −1609 | 1626 | 54  | −7 | −11 | 12
17  | 1 | 2 | 2 | 55  | 1 | 3 | 3
18  | −1 | −2 | 3 | 56  | −11 | −21 | 22
19  | 0 | −2 | 3 | 57  | 1 | −2 | 4
20  | 1 | −2 | 3 | 60  | −1 | −4 | 5
21  | −11 | −14 | 16 | 61  | 0 | −4 | 5
24  | −2901096694 | −15550555555 | 15584139827 | 62  | 2 | 3 | 3
25  | −1 | −1 | 3 | 63  | 0 | −1 | 4
26  | 0 | −1 | 3 | 64  | −3 | −5 | 6
27  | −4 | −5 | 6 | 65  | 0 | 1 | 4
28  | 0 | 1 | 3 | 66  | 1 | 1 | 4
29  | 1 | 1 | 3 | 69  | 2 | −4 | 5
30  | −283059965 | −2218888517 | 2220422932 | 70  | 11 | 20 | −21
33  | −2736111468807040 | −8778405442862239 | 8866128975287528 | 71  | −1 | 2 | 4
34  | −1 | 2 | 3 | 72  | 7 | 9 | −10
35  | 0 | 2 | 3 | 73  | 1 | 2 | 4
36  | 1 | 2 | 3 | 74  | 66229832190556 | 283450105697727 | −284650292555885
37  | 0 | −3 | 4 | 75  | 4381159 | 435203083 | −435203231
38  | 1 | −3 | 4 | 78  | 26 | 53 | −55

### Fermat's Last Theorem for cubes

[edit]

Main article: Fermat's Last Theorem

The equation x3 \+ y3 = z3 has no non-trivial (i.e. xyz ≠ 0) solutions in integers. In fact, it has none in Eisenstein integers.[6]

Both of these statements are also true for the equation[7] x3 \+ y3 = 3z3.

### Sum of first n cubes

[edit]

The sum of the first n cubes is the nth triangle number squared:

1 3 + 2 3 + ⋯ + n 3 = ( 1 + 2 + ⋯ + n ) 2 = ( n ( n + 1 ) 2 ) 2 . {\displaystyle 1^{3}+2^{3}+\dots +n^{3}=(1+2+\dots +n)^{2}=\left({\frac {n(n+1)}{2}}\right)^{2}.}
Visual proof that 13 \+ 23 \+ 33 \+ 43 \+ 53 = (1 + 2 + 3 + 4 + 5)2.

Proofs. Charles Wheatstone (1854) gives a particularly simple derivation, by expanding each cube in the sum into a set of consecutive odd numbers. He begins by giving the identity

n 3 = ( n 2 − n + 1 ) + ( n 2 − n + 1 + 2 ) + ( n 2 − n + 1 + 4 ) + ⋯ + ( n 2 + n − 1 ) ⏟ n consecutive odd numbers . {\displaystyle n^{3}=\underbrace {\left(n^{2}-n+1\right)+\left(n^{2}-n+1+2\right)+\left(n^{2}-n+1+4\right)+\cdots +\left(n^{2}+n-1\right)} _{n{\text{ consecutive odd numbers}}}.}

That identity is related to triangular numbers T n {\displaystyle T_{n}} in the following way:

n 3 = ∑ k = T n − 1 + 1 T n ( 2 k − 1 ) , {\displaystyle n^{3}=\sum _{k=T_{n-1}+1}^{T_{n}}(2k-1),}

and thus the summands forming  n 3 {\displaystyle n^{3}} start off just after those forming all previous values  1 3 {\displaystyle 1^{3}} up to  ( n − 1 ) 3 {\displaystyle (n-1)^{3}} . Applying this property, along with another well-known identity:

n 2 = ∑ k = 1 n ( 2 k − 1 ) , {\displaystyle n^{2}=\sum _{k=1}^{n}(2k-1),}

we obtain the following derivation:

∑ k = 1 n k 3 = 1 + 8 + 27 + 64 + ⋯ + n 3 = 1 ⏟ 1 3 + 3 + 5 ⏟ 2 3 + 7 + 9 + 11 ⏟ 3 3 + 13 + 15 + 17 + 19 ⏟ 4 3 + ⋯ + ( n 2 − n + 1 ) + ⋯ + ( n 2 + n − 1 ) ⏟ n 3 = 1 ⏟ 1 2 + 3 ⏟ 2 2 + 5 ⏟ 3 2 + ⋯ + ( n 2 + n − 1 ) ⏟ ( n 2 + n 2 ) 2 = ( 1 + 2 + ⋯ + n ) 2 = ( ∑ k = 1 n k ) 2 . {\displaystyle {\begin{aligned}\sum _{k=1}^{n}k^{3}&=1+8+27+64+\cdots +n^{3}\\\&=\underbrace {1} _{1^{3}}+\underbrace {3+5} _{2^{3}}+\underbrace {7+9+11} _{3^{3}}+\underbrace {13+15+17+19} _{4^{3}}+\cdots +\underbrace {\left(n^{2}-n+1\right)+\cdots +\left(n^{2}+n-1\right)} _{n^{3}}\\\&=\underbrace {\underbrace {\underbrace {\underbrace {1} _{1^{2}}+3} _{2^{2}}+5} _{3^{2}}+\cdots +\left(n^{2}+n-1\right)} _{\left({\frac {n^{2}+n}{2}}\right)^{2}}\\\&=(1+2+\cdots +n)^{2}\\\&={\bigg (}\sum _{k=1}^{n}k{\bigg )}^{2}.\end{aligned}}}
Visual demonstration that the square of a triangular number equals a sum of cubes.

In the more recent mathematical literature, Stein (1971) uses the rectangle-counting interpretation of these numbers to form a geometric proof of the identity (see also Benjamin, Quinn & Wurtz 2006); he observes that it may also be proved easily (but uninformatively) by induction, and states that Toeplitz (1963) provides "an interesting old Arabic proof". Kanim (2004) provides a purely visual proof, Benjamin & Orrison (2002) provide two additional proofs, and Nelsen (1993) gives seven geometric proofs.

For example, the sum of the first 5 cubes is the square of the 5th triangular number,

1 3 + 2 3 + 3 3 + 4 3 + 5 3 = 15 2 {\displaystyle 1^{3}+2^{3}+3^{3}+4^{3}+5^{3}=15^{2}}

A similar result can be given for the sum of the first y odd cubes,

1 3 + 3 3 + ⋯ + ( 2 y − 1 ) 3 = ( x y ) 2 {\displaystyle 1^{3}+3^{3}+\dots +(2y-1)^{3}=(xy)^{2}}

but x, y must satisfy the negative Pell equation x2 − 2y2 = −1. For example, for y = 5 and 29, then,

1 3 + 3 3 + ⋯ + 9 3 = ( 7 ⋅ 5 ) 2 {\displaystyle 1^{3}+3^{3}+\dots +9^{3}=(7\cdot 5)^{2}}

1 3 + 3 3 + ⋯ + 57 3 = ( 41 ⋅ 29 ) 2 {\displaystyle 1^{3}+3^{3}+\dots +57^{3}=(41\cdot 29)^{2}}

and so on. Also, every even perfect number, except the lowest, is the sum of the first 2⁠p−1/2⁠
odd cubes (p = 3, 5, 7, ...):

28 = 2 2 ( 2 3 − 1 ) = 1 3 + 3 3 {\displaystyle 28=2^{2}(2^{3}-1)=1^{3}+3^{3}}
496 = 2 4 ( 2 5 − 1 ) = 1 3 + 3 3 + 5 3 + 7 3 {\displaystyle 496=2^{4}(2^{5}-1)=1^{3}+3^{3}+5^{3}+7^{3}}
8128 = 2 6 ( 2 7 − 1 ) = 1 3 + 3 3 + 5 3 + 7 3 + 9 3 + 11 3 + 13 3 + 15 3 {\displaystyle 8128=2^{6}(2^{7}-1)=1^{3}+3^{3}+5^{3}+7^{3}+9^{3}+11^{3}+13^{3}+15^{3}}

### Sum of cubes of numbers in arithmetic progression

[edit]

One interpretation of Plato's number, 33 \+ 43 \+ 53 = 63

There are examples of cubes of numbers in arithmetic progression whose sum is a cube:

3 3 + 4 3 + 5 3 = 6 3 {\displaystyle 3^{3}+4^{3}+5^{3}=6^{3}}
11 3 + 12 3 + 13 3 + 14 3 = 20 3 {\displaystyle 11^{3}+12^{3}+13^{3}+14^{3}=20^{3}}
31 3 + 33 3 + 35 3 + 37 3 + 39 3 + 41 3 = 66 3 {\displaystyle 31^{3}+33^{3}+35^{3}+37^{3}+39^{3}+41^{3}=66^{3}}

with the first one sometimes identified as the mysterious Plato's number. The formula F for finding the sum of n cubes of numbers in arithmetic progression with common difference d and initial cube a3,

F ( d , a , n ) = a 3 + ( a + d ) 3 + ( a + 2 d ) 3 + ⋯ + ( a + d n − d ) 3 {\displaystyle F(d,a,n)=a^{3}+(a+d)^{3}+(a+2d)^{3}+\cdots +(a+dn-d)^{3}}

is given by

F ( d , a , n ) = ( n / 4 ) ( 2 a − d + d n ) ( 2 a 2 − 2 a d + 2 a d n − d 2 n + d 2 n 2 ) {\displaystyle F(d,a,n)=(n/4)(2a-d+dn)(2a^{2}-2ad+2adn-d^{2}n+d^{2}n^{2})}

A parametric solution to

F ( d , a , n ) = y 3 {\displaystyle F(d,a,n)=y^{3}}

is known for the special case of d = 1, or consecutive cubes, as found by Pagliani in 1829.[8]

### Cubes as sums of successive odd integers

[edit]

In the sequence of odd integers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ..., the first one is a cube (1 = 13); the sum of the next two is the next cube (3 + 5 = 23); the sum of the next three is the next cube (7 + 9 + 11 = 33); and so forth.

### Waring's problem for cubes

[edit]

Main article: Waring's problem

Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes:

23 = 23 \+ 23 \+ 13 \+ 13 \+ 13 \+ 13 \+ 13 \+ 13 \+ 13.

## In rational numbers

[edit]

Every positive rational number is the sum of three positive rational cubes,[9] and there are rationals that are not the sum of two rational cubes.[10]

## In real numbers, other fields, and rings

[edit]

Further information: cubic function

y = x3 plotted on a Cartesian plane

In real numbers, the cube function preserves the order: larger numbers have larger cubes. In other words, cubes (strictly) monotonically increase. Also, its codomain is the entire real line: the function x ↦ x3 : R → R is a surjection (takes all possible values). Only three numbers are equal to their own cubes: −1, 0, and 1. If −1 < x < 0 or 1 < x, then x3 > x. If x < −1 or 0 < x < 1, then x3 < x. All aforementioned properties pertain also to any higher odd power (x5, x7, ...) of real numbers. Equalities and inequalities are also true in any ordered ring.

Volumes of similar Euclidean solids are related as cubes of their linear sizes.

In complex numbers, the cube of a purely imaginary number is also purely imaginary. For example, i3 = −i.

The derivative of x3 equals 3x2.

Cubes occasionally have the surjective property in other fields, such as in Fp for such prime p that p ≠ 1 (mod 3),[11] but not necessarily: see the counterexample with rationals above. Also in F7 only three elements 0, ±1 are perfect cubes, of seven total. −1, 0, and 1 are perfect cubes anywhere and the only elements of a field equal to their own cubes: x3 − x = x(x − 1)(x \+ 1).

## History

[edit]

Determination of the cubes of large numbers was very common in many ancient civilizations. Mesopotamian mathematicians created cuneiform tablets with tables for calculating cubes and cube roots by the Old Babylonian period (20th to 16th centuries BC).[12][13] Cubic equations were known to the ancient Greek mathematician Diophantus.[14] Hero of Alexandria devised a method for calculating cube roots in the 1st century CE.[15] Methods for solving cubic equations and extracting cube roots appear in The Nine Chapters on the Mathematical Art, a Chinese mathematical text compiled around the 2nd century BCE and commented on by Liu Hui in the 3rd century CE.[16]

## See also

[edit]

* Cabtaxi number
* Cubic equation
* Doubling the cube
* Eighth power
* Euler's sum of powers conjecture
* Fifth power
* Fourth power
* Kepler's laws of planetary motion#Third law
* Monkey saddle
* Perfect power
* Seventh power
* Sixth power
* Square
* Taxicab number

## Notes

[edit]

1. ^ The Unicode superscript character ³ is also available for typesetting: n³.

## References

[edit]

1. ^ Huisman, Sander G. (27 Apr 2016). "Newer sums of three cubes". arXiv:1604.07746 [math.NT].
2. ^ Booker, Andrew R.; Sutherland, Andrew V. (2021). "On a question of Mordell". Proceedings of the National Academy of Sciences. 118 (11). arXiv:2007.01209. Bibcode:2021PNAS..11822377B. doi:10.1073/pnas.2022377118. PMC 7980389. PMID 33692126.
3. ^ Sequences A060465, A060466 and A060467 in OEIS
4. ^ Threecubes
5. ^ n=x^3+y^3+z^3
6. ^ Hardy & Wright, Thm. 227
7. ^ Hardy & Wright, Thm. 232
8. ^ Bennett, Michael A.; Patel, Vandita; Siksek, Samir (2017), "Perfect powers that are sums of consecutive cubes", Mathematika, 63 (1): 230–249, arXiv:1603.08901, doi:10.1112/S0025579316000231, MR 3610012
9. ^ Hardy & Wright, Thm. 234
10. ^ Hardy & Wright, Thm. 233
11. ^ The multiplicative group of Fp is cyclic of order p − 1, and if it is not divisible by 3, then cubes define a group automorphism.
12. ^ Cooke, Roger (8 November 2012). The History of Mathematics. John Wiley & Sons. p. 63\. ISBN 978-1-118-46029-0.
13. ^ Nemet-Nejat, Karen Rhea (1998). Daily Life in Ancient Mesopotamia. Greenwood Publishing Group. p. 306. ISBN 978-0-313-29497-6.
14. ^ Van der Waerden, Geometry and Algebra of Ancient Civilizations, chapter 4, Zurich 1983 ISBN 0-387-12159-5
15. ^ Smyly, J. Gilbart (1920). "Heron's Formula for Cube Root". Hermathena. 19 (42). Trinity College Dublin: 64–67\. JSTOR 23037103.
16. ^ Crossley, John; W.-C. Lun, Anthony (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford University Press. pp. 176, 213. ISBN 978-0-19-853936-0.

### Sources

[edit]

* Benjamin, Arthur T.; Orrison, Michael E. (November 2002). "Two Quick Combinatorial Proofs of Σ k = 1 n k 3 = ( \smallmatrix n+1 2 \endsmallmatrix ) 2" (PDF). The College Mathematics Journal. 33 (5): 406. doi:10.2307/1559017. JSTOR 1559017.
* Benjamin, Arthur T.; Quinn, Jennifer J.; Wurtz, Calyssa (1 November 2006). "Summing Cubes by Counting Rectangles". The College Mathematics Journal. 37 (5): 387–389\. doi:10.2307/27646391. JSTOR 27646391.
* Hardy, G. H.; Wright, E. M. (1980). An Introduction to the Theory of Numbers (Fifth ed.). Oxford: Oxford University Press. ISBN 978-0-19-853171-5.
* Kanim, Katherine (1 October 2004). "Proof without Words: The Sum of Cubes: An Extension of Archimedes' Sum of Squares". Mathematics Magazine. 77 (4): 298–299\. doi:10.2307/3219288. JSTOR 3219288.
* Nelsen, Roger B. (1993). Proofs without words : exercises in visual thinking. Cambridge University Press. ISBN 978-0-88385-700-7.
* Stein, Robert G. (1 May 1971). "A Combinatorial Proof That Σ k3 = (Σ k)2". Mathematics Magazine. 44 (3): 161–162\. doi:10.2307/2688231. JSTOR 2688231.
* Toeplitz, Otto (1963). The calculus: a genetic approach. Chicago: University of Chicago Press. ISBN 978-0-226-80667-9. `{{cite book}}`: ISBN / Date incompatibility (help)
* Wheatstone, C. (1854). "On the formation of powers from arithmetical progressions". Proceedings of the Royal Society of London. 7: 145–151\. Bibcode:1854RSPS....7..145W. doi:10.1098/rspl.1854.0036. S2CID 121885197.

* v
* t
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Figurate numbers

2-dimensional|

| centered|

* Centered triangular numbers
* Centered square numbers
* Centered pentagonal numbers
* Centered hexagonal numbers
* Centered heptagonal numbers
* Centered octagonal numbers
* Centered nonagonal numbers
* Centered decagonal numbers
* Star numbers

|

non-centered|

* Triangular numbers
* Square numbers
* Pentagonal numbers
* Hexagonal numbers
* Heptagonal numbers
* Octagonal numbers
* Nonagonal numbers
* Decagonal numbers
* Dodecagonal numbers

3-dimensional|

| centered|

* Centered tetrahedral numbers
* Centered cube numbers
* Centered octahedral numbers
* Centered dodecahedral numbers
* Centered icosahedral numbers

|

non-centered|

* Cube numbers
* Octahedral numbers
* Dodecahedral numbers
* Icosahedral numbers
* Stella octangula numbers

pyramidal|

* Tetrahedral numbers
* Square pyramidal numbers

4-dimensional|

| non-centered|

* Pentatope numbers
* Squared triangular numbers
* Tesseractic numbers

|

Higher dimensional|

| non-centered|

* 5-hypercube numbers
* 6-hypercube numbers
* 7-hypercube numbers
* 8-hypercube numbers

|

* v
* t
* e

Classes of natural numbers

Powers and related numbers

* Achilles
* Power of 2
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* Power of 10
* Square
* Cube
* Fourth power
* Fifth power
* Sixth power
* Seventh power
* Eighth power
* Perfect power
* Powerful
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Of the form a × 2b ± 1

* Cullen
* Double Mersenne
* Fermat
* Mersenne
* Proth
* Thabit
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Other polynomial numbers

* Hilbert
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* Lucky numbers of Euler

Recursively defined numbers

* Fibonacci
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Possessing a specific set of other numbers

* Amenable
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Expressible via specific sums

* Nonhypotenuse
* Polite
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* Ulam
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Figurate numbers

| 2-dimensional|

| centered|

* Centered triangular
* Centered square
* Centered pentagonal
* Centered hexagonal
* Centered heptagonal
* Centered octagonal
* Centered nonagonal
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* Star

|

non-centered|

* Triangular
* Square
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* Pentagonal
* Hexagonal
* Heptagonal
* Octagonal
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* Decagonal
* Dodecagonal

3-dimensional|

| centered|

* Centered tetrahedral
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* Centered octahedral
* Centered dodecahedral
* Centered icosahedral

|

non-centered|

* Tetrahedral
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* Dodecahedral
* Icosahedral
* Stella octangula

pyramidal|

* Square pyramidal

4-dimensional|

| non-centered|

* Pentatope
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* Tesseractic

|

Combinatorial numbers

* Bell
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* Lazy caterer's sequence
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* Telephone number
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Primes

* Wieferich
* Wall–Sun–Sun
* Wolstenholme prime
* Wilson

Pseudoprimes

* Carmichael number
* Catalan pseudoprime
* Elliptic pseudoprime
* Euler pseudoprime
* Euler–Jacobi pseudoprime
* Fermat pseudoprime
* Frobenius pseudoprime
* Lucas pseudoprime
* Lucas–Carmichael number
* Perrin pseudoprime
* Somer–Lucas pseudoprime
* Strong pseudoprime

Arithmetic functions and dynamics

| Divisor functions|

* Abundant
* Almost perfect
* Arithmetic
* Betrothed
* Colossally abundant
* Deficient
* Descartes
* Hemiperfect
* Highly abundant
* Highly composite
* Hyperperfect
* Multiply perfect
* Perfect
* Practical
* Primitive abundant
* Quasiperfect
* Refactorable
* Semiperfect
* Sublime
* Superabundant
* Superior highly composite
* Superperfect

|

Prime omega functions|

* Almost prime
* Semiprime

Euler's totient function|

* Highly cototient
* Highly totient
* Noncototient
* Nontotient
* Perfect totient
* Sparsely totient

Aliquot sequences|

* Amicable
* Perfect
* Sociable
* Untouchable

Primorial|

* Euclid
* Fortunate

Other prime factor or divisor related numbers

* Blum
* Cyclic
* Erdős–Nicolas
* Erdős–Woods
* Friendly
* Giuga
* Harmonic divisor
* Jordan–Pólya
* Lucas–Carmichael
* Pronic
* Regular
* Rough
* Smooth
* Sphenic
* Størmer
* Super-Poulet

Numeral system-dependent numbers

| Arithmetic functions
and dynamics|

* Persistence
* Additive
* Multiplicative

| Digit sum|

* Digit sum
* Digital root
* Self
* Sum-product

|

Digit product|

* Multiplicative digital root
* Sum-product

Coding-related|

* Meertens

Other|

* Dudeney
* Factorion
* Kaprekar
* Kaprekar's constant
* Keith
* Lychrel
* Narcissistic
* Perfect digit-to-digit invariant
* Perfect digital invariant
* Happy

P-adic numbers-related|

* Automorphic
* Trimorphic

Digit-composition related|

* Palindromic
* Pandigital
* Repdigit
* Repunit
* Self-descriptive
* Smarandache–Wellin
* Undulating

Digit-permutation related|

* Cyclic
* Digit-reassembly
* Parasitic
* Primeval
* Transposable

Divisor-related|

* Equidigital
* Extravagant
* Frugal
* Harshad
* Polydivisible
* Smith
* Vampire

Other|

* Friedman

Binary numbers

* Evil
* Odious
* Pernicious

Generated via a sieve

* Lucky
* Prime

Sorting related

* Pancake number
* Sorting number

Natural language related

* Aronson's sequence
* Ban

Graphemics related

* Strobogrammatic

* Mathematics portal

* v
* t
* e

Sequences and series

Integer sequences|

| Basic|

* Arithmetic progression
* Geometric progression
* Harmonic progression
* Square number
* Cubic number
* Factorial
* Powers of two
* Powers of three
* Powers of 10

|

Advanced (list)|

* Complete sequence
* Fibonacci sequence
* Figurate number
* Heptagonal number
* Hexagonal number
* Lucas number
* Pell number
* Pentagonal number
* Polygonal number
* Triangular number
* array

Properties of sequences|

* Cauchy sequence
* Monotonic function
* Periodic sequence

Properties of series|

| Series|

* Alternating
* Convergent
* Divergent
* Telescoping

|

Convergence|

* Absolute
* Conditional
* Uniform

Explicit series|

| Convergent|

* 1/2 − 1/4 + 1/8 − 1/16 + ⋯
* 1/2 + 1/4 + 1/8 + 1/16 + ⋯
* 1/4 + 1/16 + 1/64 + 1/256 + ⋯
* 1 + 1/2s \+ 1/3s \+ ... (Riemann zeta function)

|

Divergent|

* 1 + 1 + 1 + 1 + ⋯
* 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
* 1 + 2 + 3 + 4 + ⋯
* 1 − 2 + 3 − 4 + ⋯
* 1 + 2 + 4 + 8 + ⋯
* 1 − 2 + 4 − 8 + ⋯
* Infinite arithmetic series
* 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
* 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
* 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)

Kinds of series|

* Taylor series
* Power series
* Formal power series
* Laurent series
* Puiseux series
* Dirichlet series
* Trigonometric series
* Fourier series
* Generating series

Hypergeometric series|

* Generalized hypergeometric series
* Hypergeometric function of a matrix argument
* Lauricella hypergeometric series
* Modular hypergeometric series
* Riemann's differential equation
* Theta hypergeometric series

* Category

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