Square numbers and cube numbers
Find square numbers here as a list with differences, plus cube numbers, triangular numbers and other figurate numbers. You can also check whether a number belongs: 144 is 12 × 12.
Source: Wikipedia – Quadratzahl (square number: sequence of squares, odd differences, last digits). Updated: .
How it is calculated
A square number is a whole number multiplied by itself: 1, 4, 9, 16, 25, 36 … The twelfth is 12 × 12 = 144. Cube numbers are n × n × n: 1, 8, 27, 64, 125.
Patterns in the sequence
The gap between two square numbers is always an odd number: 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7. That is why the sum of the first n odd numbers is exactly n². Square numbers end only in 0, 1, 4, 5, 6 or 9. Two consecutive triangular numbers always add up to a square: 6 + 10 = 16.
Other figurate numbers
Triangular numbers n(n+1)/2 count dots in a triangle (1, 3, 6, 10, 15), pentagonal numbers n(3n−1)/2 in a pentagon, hexagonal numbers n(2n−1) in a hexagon and tetrahedral numbers n(n+1)(n+2)/6 in a pyramid.
- Checking: the calculator finds n by bisection and names the neighbours of the number.
- For square numbers it shows the integer root with remainder: √150 ≈ 12 remainder 6.
Frequently asked questions
Which numbers are square numbers?
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 … that is n × n for n = 1, 2, 3 …
How can I tell whether a number is a square?
It ends in 0, 1, 4, 5, 6 or 9. Only the root test is certain: if the root is a whole number, the number is a square.
What are cube numbers?
The third powers of whole numbers: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
Sources and legal basis
- Wikipedia – Quadratzahl (square number: sequence of squares, odd differences, last digits)
- OEIS A000290 – The squares: a(n) = n^2
- Wikipedia – Figurierte Zahl (triangular, pentagonal, hexagonal and tetrahedral numbers)
- OEIS A000217 – Triangular numbers: a(n) = n(n+1)/2
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