# Every integer is interesting!

Proposition: Every integer has an interesting property that can be described in 19 words or less.

Proof by contradiction: Suppose that there exists numbers which do not have an interesting property. Let $S$ be the smallest of these numbers by the Well-Ordering Principle. Then,

"S is the smallest integer that cannot be described in 14 words or less."

The point of this note is to list out an interesting property for each positive integer. Reply to the largest number N, and state why N+1 is interesting in 14 words or less.

Rules:
2. Make sure you use 14 words or less.
3. Do not reply out of sequence.
4. Do not reply to your own comment. (Applicable to 9 onwards)

Note by Calvin Lin
5 years, 3 months ago

1 vote

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MarkdownAppears as
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Note: you must add a full line of space before and after lists for them to show up correctly
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MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
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\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

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1 is the only positive integer that is neither prime nor composite.

Staff - 5 years, 3 months ago

2 is the only even prime number

- 5 years, 3 months ago

3 is the only prime that is 1 less than a perfect square

- 5 years, 3 months ago

4 is the maximum number of dimensions of a real division algebra

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5 is the second prime in the first pair of twin primes!

- 5 years, 3 months ago

6 is the first perfect number!

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7 is the number of all possible Frieze groups

- 5 years, 3 months ago

8 is the first cube of a prime!

- 5 years, 3 months ago

9 is the maximum number of cubes required to sum to any integer

- 5 years, 3 months ago

10 is the base that humans use to count.

Staff - 5 years, 3 months ago

11 is the smallest integer palindrome

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12 is the number of possible 3x3 latin squares!

- 5 years, 3 months ago

13 is the number of all possible Archimedian solids

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14 is the sum of the first three square numbers!

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15 is the smallest magic constant of any non-trivial magic square

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16 is the first non-trivial Erdős-Woods number!

- 5 years, 3 months ago

After 5, 17 is least prime number of sides of a constructible regular polygon

- 5 years, 3 months ago

18 is the only positive integer that is twice the sum of it's digits!

- 5 years, 3 months ago

19 is the smallest prime that is a different prime when turned upsidedown

19 is the world's record of number of beer steins carried by a barmaid

19 is the date of my birthday, and I'm out of town now, carry on!

- 5 years, 3 months ago

20 is the smallest primitive abundant number and least 2-digit magic number.

- 5 years, 3 months ago

21 is the smallest, nontrivial fibonacci number whose digit sum is a fibonacci number!

- 5 years, 3 months ago

22 is the smallest number expressible as the sum of 2 primes in 3 ways.

- 5 years, 3 months ago

23 is the smallest odd prime that has no twin prime :D -> (14 words or less).

23 has the distinction of being only one of two integers (other is 239) that cannot be expressed as the sum of fewer than 9 cubes of integers. See Waring's Problem -> (14 words or more)

- 5 years, 3 months ago

24 is the largest integer divisible by all positive integers less than its square root.

- 5 years, 3 months ago

25 is the smallest square that is also a sum of two (non-zero) distinct squares (9 and 16).

- 5 years, 3 months ago

26 is the only integer that is one greater than a square (25 = 5^2) and one less than a cube (27 = 3^3)

- 5 years, 3 months ago

27 is the only positive integer that is 3 times the sum of its digits.

In base 10, 27 is the first composite number not evenly divisible by any of its digits.

27 contains the decimal digits 2 and 7, and is the result of adding together the integers from 2 to 7 (2+3+4+5+6+7=27).

In the Collatz conjecture, a starting value of 27 requires 112 steps to reach 1, many more than any lower number.

- 5 years, 3 months ago

28 is the sum of the totient function for the first nine integers

- 5 years, 3 months ago

29 is the sum of three consecutive squares!

- 5 years, 3 months ago

30 is the smallest sphenic number.

- 5 years, 3 months ago

31 is a centered triangular number, a centered pentagonal number and centered decagonal number.

- 5 years, 3 months ago

32 is the ninth happy number.

- 5 years, 3 months ago

33 is the smallest sum of two different positive numbers, each of which raised to the fifth power.

33 is the largest positive integer that cannot be expressed as a sum of different triangular numbers.

33 is the smallest integer such that it and the next two integers all have the same number of divisors.

- 5 years, 3 months ago

34 is the ninth Fibonacci number

- 5 years, 3 months ago

35 is the sum of the first five triangular numbers, making it a tetrahedral number.

35 is the highest number one can count to on one's fingers using base 6.

- 5 years, 3 months ago

36 is the square of the first perfect number.

- 5 years, 3 months ago

37 - Every positive integer is the sum of at most 37 fifth powers. See Waring's Problem.

- 5 years, 3 months ago

38 is the sum of the squares of the first three primes.

Largest even number which cannot be written as sum of two odd composite numbers.

- 5 years, 3 months ago

39 is the smallest natural number which has three partitions into three parts which all give the same product when multiplied: {25, 8, 6}, {24, 10, 5}, {20, 15, 4}.

- 5 years, 3 months ago

40 is the only number whose (English) spelling consists of letters in alphabetical order.

- 5 years, 3 months ago

41 is the smallest integer whose reciprocal has a 5-digit repetend.

- 5 years, 3 months ago

42 is The Answer to The Ultimate Question of Life, The Universe and Everything.

- 5 years, 3 months ago

43 is the smallest prime number expressible as the sum of 2, 3, 4, or 5 different primes.

*43 = 41 + 2

*43 = 11 + 13 + 19

*43 = 2 + 11 + 13 + 17

*43 = 3 + 5 + 7 + 11 + 17.

- 5 years, 3 months ago

43 is the first centered heptagonal prime.

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44 is the number of derangements of 5 distinct objects.

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45 is the smallest triangular number that is a sum of two positive squares.

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46 is the number of chromosomes in every cell of the human body.

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47 - There are 47 occurrences of '47' in the first thousand prime numbers. .

$\large{\dfrac{9999999999999999999999999999999999999999999999}{212765957446808510638297872340425531914893617} = 47}$ .

47 is the largest number of cubes that cannot tile a cube.

- 5 years, 3 months ago

48 is the smallest natural number with exactly ten positive factors.

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49 is expressible as a sum of three different primes in eight different ways.

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50 is the number of bonus points earned by using seven tiles in Scrabble.

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51 is the least composite shortest side length of an almost-equilateral Heronian triangle.

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52 is the number of playing cards in a standard deck.

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The sum of the first 53 primes is divisible by 53.

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54 is the smallest number having 3 representations as sum of 3 positive squares.

- 5 years, 3 months ago

55 is the largest Fibonacci number that is triangular.

- 5 years, 3 months ago

56 is the sum of the first six triangular numbers (making it a tetrahedral number), as well as the sum of first six odd consecutive primes (3 + 5 + 7 + 11 + 13 + 17).

- 5 years, 3 months ago

57 is the number of strokes that the most complex Chinese character has.

- 5 years, 3 months ago

58 is the sum of first 7 prime numbers.

- 5 years, 3 months ago

There are 59 stellations of the icosahedron.

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There are 60 one-sided hexominoes.

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61 is the smallest prime whose square is expressible as the sum of 1,2,3 and 4 distinct squares.

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62 - Only number whose cube consists of 3 digits each occurring two times.

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63 is the smallest positive integer $N$ such that $\frac{N}{k}, 1 \le k \le 9$ has no repeating decimals.

- 5 years, 3 months ago

There were 64 golden discs in the myth of Tower of Hanoi.

- 5 years, 3 months ago

65 is equal to $1^{5} + 2^{4} + 3^{3} + 4^{2} + 5^{1}.$

- 5 years, 3 months ago

66 is the first sphenic triangular number.

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67 is the number of counties present in each of Florida and Pennsylvania.

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On an infinite chessboard, a knight can reach 68 different squares after three moves.

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69 is an integer such that its square and cube concatenated contains each of $0$ through $9$ exactly once.

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70 is the smallest weird number.

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71 is conjectured to be the largest number whose square is a factorial plus one, ($71^{2} = 7! + 1$).

- 5 years, 3 months ago

72: Smallest number whose fifth power is the sum of five smaller fifth powers $(19^5 + 43^5 + 46^5 + 47^5 + 67^5 = 72^5)$.

- 5 years, 3 months ago

73 is the smallest positive integer $n$ such that each of $n - 1, n$ and $n + 1$ are expressible as sums of two squares.

- 5 years, 3 months ago

74 is the number of different non-Hamiltonian polyhedra with a minimum number of vertices.

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75 is the number of orderings of 4 distinct objects with ties allowed.

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76 is an automorphic number because ${75}^{2}$'s last 2 digits is 75.

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77 is the sum of first 8 prime numbers.

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78 is the number of lines that make up Metatron's Cube

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79: Smallest number that can't be represented as sum of fewer than 19 fourth powers.

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80:The sum of Euler's totient function φ(x) over the first sixteen integers is 80.

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81 is the square of the sum of its digits.

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82:a happy number

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83 is the number of strongly connected digraphs with 4 vertices.

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A hepteract has 84 penteract 5-faces.

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85 is the largest $n$ for which \sum _{ x=1 }^{ n }{ { x }^{ 2 } } =\sum _{ y=1 }^{ m }{ y } for some $m$.

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$2^{86}$ is conjectured to be the highest power of 2 without a 0 in its decimal expansion.

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87 is the sum of the squares of the first 4 primes.

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88 is conjectured to be the only number whose square has no "isolated" digits. ($88^{2} = 7744.$)

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89 is the smallest Sophie Germain prime to start a Cunningham chain of the first kind of six terms.

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90 is the sum of its digits plus the squares of its digits.

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91 is the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed (alternatively the sum of two cubes and the difference of two cubes): $91 = 6^3+(-5)^3 = 4^3+3^3$

91 is the smallest positive integer expressible as a sum of six distinct squares: $91 = 1^2+2^2+3^2+4^2+5^2+6^2$

91 is the smallest pseudoprime satisfying the congruence $3^n \equiv 3 \pmod{n}$.

+91 is the international direct dial phone call code for India.

- 5 years, 3 months ago

There are 92 naturally occurring elements on earth.

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There are 93 different cyclic Gilbreath permutations on 11 elements, and therefore there are 93 different real periodic points of order 11 on the Mandelbrot set.

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94 is the second number in the third triplet of three consecutive distinct semiprimes, 93, 94 and 95.

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95 is a Thabit number.

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Every integer greater than 96 may be represented as a sum of distinct super-prime numbers.

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97 is the largest two-digit prime.

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98 is not in the range of $\phi$.

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99 is the sum of the cubes of three consecutive integers:

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100 is the smallest three-digit number.

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101 is the smallest three-digit prime, and also a palindrome.

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102 contains each of the first three digits exactly once.

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103 is the smallest cyclops prime with distinct digits

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104 is the first three digit product of two consecutive Fibonacci numbers

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105 is the smallest dodecagonal number that is also a double factorial

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106 is the only centered pentagonal and heptagonal number that differ by one layer

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106 is the number of distinct trees with ten vertices.

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107 is the smallest three digit Mersenne prime exponent

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108 is the interior angle of the regular pentagon

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109 is the smallest number that has more different digits that its square

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110 is the smallest pronic number that is the sum of three consecutive squares

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111 is the smallest repunit semiprime

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112 is the smallest heptagonal number that is the sum of six consecutive primes

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113 is the smallest three digit permutable prime

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114 is the sum of the first four hyperfactorials

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115 is the smallest odd heptagonal pyramidal semiprime

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116 is the actual number of years the Hundred Years' War lasted.

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117 is the smallest pentagonal number divisible by an odd square

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118 is smallest three digit number leaving remainders (1,2,3,4) when divided by (3,4,5,6)

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119 is the smallest "prime-looking" number greater than 100.

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120 is $5!$

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121 is the smallest nontrivial palindromic square.

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76 raise to power any positive integer end with 76

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30 is the sum of first four squares.

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29 is the smallest two digit primorial prime, which forms a twin prime with another primorial prime (31).

- 5 years, 3 months ago
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