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78
78 is an integer, the smallest number that can be written as the sum of of four distinct squares in three different ways: 8 2 + 3 2 + 2 2 + 1 2 = 7 2 + 4 2 + 3 2 + 2 2 = 6 2 + 5 2 + 4 2 + 1 2 = 78.
Contents
- 1 Membership in core sequences
- 2 Sequences pertaining to 78
- 3 Partitions of 78
- 4 Roots and powers of 78
- 5 Values for number theoretic functions with 78 as an argument
- 6 Factorization of some small integers in a quadratic integer ring adjoining the square roots of −78, 78
- 7 Factorization of 78 in some quadratic integer rings
- 8 Representation of 78 in various bases
- 9 See also
Membership in core sequences
Even numbers | ..., 72, 74, 76, 78, 80, 82, 84, ... | A005843 |
Composite numbers | ..., 75, 76, 77, 78, 80, 81, 82, ... | A002808 |
Triangular numbers | ..., 36, 45, 55, 66, 78, 91, 105, ... | A000217 |
Squarefree numbers | ..., 73, 74, 77, 78, 79, 82, 83, ... | A005117 |
Abundant numbers | ..., 66, 70, 72, 78, 80, 84, 88, ... | A005101 |
Sequences pertaining to 78
Multiples of 78 | 0, 78, 156, 234, 312, 390, 468, 546, 624, 702, 780, 858, 936, ... | |
Divisors of 78 | 1, 2, 3, 6, 13, 26, 39, 78 | A018274 |
sequence beginning at 78 | 78, 39, 118, 59, 178, 89, 268, 134, 67, 202, 101, 304, 152, 76, ... |
Partitions of 78
There are 12132164 partitions of 78.
The Goldbach representations of 78 are: 73 + 5 = 71 + 7 = 67 + 11 = 61 + 17 = 59 + 19 = 47 + 31 = 41 + 37.
Roots and powers of 78
In the table below, irrational numbers are given truncated to eight decimal places.
TABLE GOES HERE
Values for number theoretic functions with 78 as an argument
TABLE GOES HERE
Factorization of some small integers in a quadratic integer ring adjoining the square roots of −78, 78
REMARKS
TABLE
To drive home the point that has class number 4, we'll show a few more numbers which not only have more than one distinct factorization, but the distinct factorizations have a different number of irreducible factors.
TABLE
Ideals really help us make sense of multiple distinct factorizations in these domains.
TABLE
Factorization of 78 in some quadratic integer rings
As was mentioned above, 78 is the product of three primes in . But it has different factorizations in some quadratic integer rings.
TABLE
REMARKS
Representation of 78 in various bases
Base | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
Representation | 1001110 | 2220 | 1032 | 303 | 210 | 141 | 116 | 86 | 78 | 71 | 66 | 60 | 58 | 53 | 4E | 4A | 46 | 42 | 3I |
See also
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 |