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78

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This article is under construction.            

Please do not rely on any information it contains.            


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.

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

Some integers
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