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Seven
Seven is the fourth-smallest prime number; the next is 11.
Seven is a Mersenne prime, since 23 - 1 = 7.
Seven is the fourth Mersenne prime exponent.
Seven is the first Newman-Shanks-Williams prime.
Seven is the third Lucas prime.
Seven is the first Woodall prime.
Seven is the second lucky prime.
Seven is the second safe prime.
Seven is the fourth factorial prime
Seven is both a Carol number and a Kynea number.
Seven is the smallest number that can't be represented as a sum of fewer than four nonzero squares.
Seven is a self number.
 Seven is the smallest happy number, except for the trivial example of 1.
Although the divisible by seven (7) test is not well known (especially compared to the popular divisible by three (3) test), there is an easy way to test if a natural number is evenly divisible by seven (7). See also Divisibility rule.
Remove the last digit,
Double it, and
Subtract it from the remaining digits.
If the result is negative and there are 2 or more digits, drop the negative sign.
Repeat until you end up with a result that is a multiple of seven (7). (i.e. -7, 0, or +7)
For example, the number 1358 is evenly divisible by seven, since:
135 - (8*2) = 119
11 - (9*2) = -7
Using Number Theory the proof is rather easy, once we rewrite the number n in the form:
n = 10a + b
Where:
a is the remaining digits, and
b is the last digit.
Then:
10a + b = 0 (mod 7)
5 * (10a + b) = 0 (mod 7)
49a + a + 5b = 0 (mod 7)
a + 5b - 7b = 0 (mod 7)
a - 2b = 0 (mod 7)
999,999 divided by 7 is exactly 142,857, so vulgar fractions with 7 in the denominator have six-digit repeating sequences in their decimal expansions. 1/7 = 0.142857142....
A seven-sided polygon is a heptagon. The regular n-gons for n = 6 can be constructed by compass and straightedge alone, but the regular heptagon cannot. Figurate numbers representing heptagons (including seven) are called heptagonal numbers. Seven is also a centered hexagonal number.
There are seven frieze groups, the groups consisting of symmetries of the plane whose group of translations is isomorphic to the group of integers.
A group or a series of seven is a heptad.
7 is the only dimension, besides the familiar 3, in which a vector cross product can be defined.
In numeral systems
| Base |
Numeral |
system |
| 2 |
binary |
111 |
| 3 |
ternary |
21 |
| 4 |
quaternary |
13 |
| 5 |
quinary |
12 |
| 6 |
senary |
11 |
| 7 |
septenary |
10 |
| over 7 |
(octal, decimal) |
7 |
In quaternary, 7 is the smallest prime with a composite sum of digits. |
