Number 15 in Math

A page of the Numeropedia - the Special Encyclopedia of Numbers

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A triangular number: 15 = 1+2+3+4+5.

The pair 15 and 21 is the smallest pair of triangular numbers whose sum (36) and difference (6) are also triangular.

Two consecutive sums of consecutive numbers: 4+5+6 = 7+8 = 15.

Product of the first 2 odd prime numbers (or first 2 Fermat prime numbers): 15 = 3×5.

A regular polygon with this odd number (product of Fermat prime numbers) of sides is constructible by using only straightedge and compass. 15 = 3´5.

The smallest composite number that there is only one finite group whose order equals that number.

154 = 44+64+84+94+144 = 50,625. The smallest 4th power that can be expressed as the sum of five 4th powers.

A base system of multigrade equalities:

1+6+8 = 15 = 2+4+9 and

12+62+82 = 101 = 22+42+92.

Multigrade equalities still hold by adding any integer n to every base term in both sides.  Hence, the base equality always has number 1 appearing in one side.

759,375 = (7–5+9–3+7)5 = 155.

Repeatedly inserting numbers 15 in the middle of the square number 16 yields other squares 1156 = 342, 111556 = 3342, 11115556 = 33342

Inserting 72 between 2 digits 2 in the square number 225 = 152 and repeatedly inserting numbers 72 between the digits 7 and 2 yields other squares 27225 = 1652, 2772225  = 16652, 277722225 = 166652, 27777222225 = 1666652, 2777772222225 = 16666652

From the most basic 3×3 magic square, where sum of 3 numbers in each row, column and main (left & right) diagonals are all 15:

 8 1 6 3 5 7 4 9 2

.   rows:           8162+3572+4922 = 1,035,369 = 6182+7532+2942

.   columns:      8342+1592+6722 = 1,172,421 = 4382+9512+2762

.   right diagonals:  8522+1742+6392 = 1,164,501 = 2582+4712+9362

and                    8522+3962+4172 = 1,056,609 = 2582+6932+7142

.   left diagonals:    6542+8792+1322 = 1,217,781 = 4562+9782+2312

and                    6542+7982+2132 = 1,109,889 = 4562+8972+3122.

The number of prime factors of the order of the Monster group M: 246×320×59×76×112×133×17×19×23×29×31×41×47×59×71 = 808017424794512875886459904961710757005754368000000000.