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# mathematics in three

3×37 = 111

33×3367 = 111,111

333×333667 = 111,111,111

3333×33336667 = 111,111,111,111

33333×3333366667 = 111,111,111,111,111

The smallest odd prime number.

A triangular number (sum of all integers from 1 to 2): 3 = 1+2.

The only triangular number that is a prime number. The only triangular number that is a Fermat number.

The only number that is the sum of the preceding integers: 3 = 1+2.

A number that is sum of factorials of the two preceding integers: 3 = 1!+2!.

The other case is 2 = 2! = 1!+0!.

The first Mersenne prime number: 22–1.

A Mersenne exponent: 23–1 = 7 is the 2nd Mersenne prime number.

Numbers 2 and 3 are the only 2 consecutive numbers, which are both prime numbers. This is the only twin prime numbers whose difference is 1.

Three numbers 3, 5 and 7 are the only three consecutive odd numbers, which are all prime numbers.

The set of 4 numbers (1, 3, 8, 120) has a property that the product of any 2 numbers is always equal a square number minus 1.

One of only 3 known square numbers, which are a factorial plus 1: 25 = 52 = 4!+1, 121 = 112 = 5!+1 and 5041 = 712 = 7!+1. Mathematician Paul ErdÃ¶s conjectured that these are the only such numbers, called Brown numbers.

The first Fermat number is a prime number. Fermat number is a number of the form (1+2^(2n)). An equilateral triangle is constructible by using only straightedge and compass.

The 4th Fibonacci number. The first and only Fibonacci number that is a prime number but its subscript is composite. There are only 4 triangular numbers that are also Fibonacci numbers: 1, 3, 21 and 55.

The 2nd Lucas number.

The first Cullen prime number, i.e. prime number of the form n´2n+1, here n = 1.

The number of pillars (or pegs) used in the Tower of Hanoi puzzle (or the Tower of Brahma or the End of World puzzle), invented by the French mathematician Edouard Lucas in 1883. Original version puzzle has 8 circular rings of decreasing sizes placed on one pillar with the largest ring at the bottom.

(3, 4, 5) is the smallest a Pythagorean triple (lengths of 3 sides of a right triangle).

Sum of all digits in a number divisible by 3 is also divisible by 3.

153,749,628 = 3×51,249,876, using each of 9 digits 1-9 once on both sides of the equality.

2×3×5×67×1489 = 2,992,890, the smallest such number, using each of 9 digits 1-9 once in its factorization.

123,456 = 643×64×3.

Sum of all prime numbers from 3 to 13 is the product of 2 numbers 3 and 13: 39 = 3×13 = 3+5+7+11+13.

285,714×3 = 857,142.

142,857×3 = 428,571.

076,923×3 = 230,769.

153,846×3 = 461,538.

230,769×3 = 692,307.

307,692×3 = 923,076.

65+56 = 112 = 121 and 65–56 = 32 = 9.

An arbitrary angle cannot be divided into 3 equal angles (or trisected) by using only straightedge and compass.

The number of sides/angles of a triangle:

3 = (5+7)/(1+3) = (7+9+11)/(1+3+5) = (9+11+13+15)/(1+3+5+7) =

= (11+13+15+17+19)/(1+3+5+7+9) = …

From the most basic 3×3 magic square:

[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+4172+3962 = 1,056,609 = 2582+7142+6932

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

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

This fact is true for any 3×3 magic square.

In a triangle,

3 perpendiculars dropped from the vertices on the opposite sides intersect at the orthocenter.

3 medians from the vertices to the midpoints of the opposite sides intersect at the centroid.

3 internal angle bisectors intersect at the incenter, which is the center of the circle tangent to its 3 sides.

3 perpendicular bisectors of three sides intersect at the circumcenter, which is the center of the circle passing through its 3 vertices.

3 symmedians intersect at the Lemoine point (or Grebe point, symmedian point). For each angle, the internal angle bisector bisects the angle formed by the symmedian and the median.

An internal angle bisector and two of the other external angle bisectors intersect at an excenter of an excircle, tangent to its 3 sides. There are 3 excircles, corresponding to 3 angles.

From a point on the circumcircle of a triangle, drop 3 perpendiculars to each side of the triangle. The 3 points of intersection all lie on a line called the Simson line.

There is exactly one circle passing through any 3 non-linear points in the 2-dimension Euclidean plane.

Each ratio 17469/5823 = 17496/5832 = 3, using each of 9 digits 1-9 once, shows how to arrange a 9-book set on 2 shelves to mark the book #3.

The only known prime whose reciprocal has period 1: 3–1 = 0.333333333… (= 0.3).

72 = 24×3 and, reversely, 27 = 24+3.

The set of 4 numbers (1, 3, 8, 120) has a property that the product of any 2 numbers is always equal a square number minus 1.

3 = 4+4–5 = 43+43–53.

In any triplets of integers for the sides of a Pythagorean triangle: 1 integer is always divisible by 3 and 1 by 5; the product of 2 legs is divisible by 12 and the product of 3 sides is divisible by 60.

The Fermat quotient (2p –1–1)/p is a square number only when p = 3 and 7.

Goldbach’s conjecture, in his letter to Leonhard Euler, dated on 7 June 1742: Every even integer (≥ 4) is the sum of 2 prime numbers. Or equivalently, every integer > 5 is the sum of 3 prime numbers.

A sphenic number is a number that has precisely 3 distinct prime factors. The first sphenic numbers are: 30, 42, 66…

The (3x+1) problem: Start with any natural number. Divide it by 2 if it is even or multiply by 3 and add 1 if it is odd. The sequence will eventually end by three numbers 4, 2, 1… repeatedly.

Eugenio Calabi’s triangle: Given any equilateral triangle with its three equal largest squares that fit inside it, there is uniquely a triangle in which there equally large squares can fit. The ratio of the largest side to the other two (equal) sides is an algebraic number x = 1.55138752455… satisfying the equation: 2x3–2x2–3x+2 = 0.

The product of the first 8 consecutive prime numbers divided by 10:

2×3×5×7×11×13×17×19/10 = 969,969 is a palindromic number.

510,510 is the product of first 7 prime numbers, of 2 consecutive numbers and of 4 consecutive Fibonacci numbers: 510,510 = 2×3×5×7×11×13×17 = 714×715 = 13×21×34×55.

3,122,490 = 2×3×5×7×14869, whose prime factors use each of 9 digits 1-9 once.

14,368,485 = 3×5×17×29×29×67, using each of 9 digits 1-9 twice.

The equality 9168×3 = 27,504 uses each of 10 digits 0-9 once.

40,578,660 = 2×2×3×3×5×17×89×149, where each digit 1-9 appears exactly twice.

A supersingular prime numbers 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.

Interesting formula: arctan1+arctan2+arctan3 = 180o.

652–562 = 332, 65+56 = 112 and 65–56 = 32.

The imaginary quadratic field Q((-3)1/2) is one of only 9 UFDs (unique factorization domains) of the formQ((-d)1/2), where d = 1, 2, 3, 7, 11, 19, 43, 67 and 163.

pi = 3.141526…

The number pi = 3.141592653589793238462… is the ratio of a circle’s circumference to its diameter.

A half revolution, a straight angle (or a semicircle) is 180o = p radians = 200 grads (grades). (Units for anglemeasurements).

[Trivia]  The “Pi Day” is celebrated on 03/14 at 1:59:26PM.