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 SS 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."

which is a contradiction.


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:
1. Start with "N is ...".
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
4 years, 4 months ago

1 vote

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Comments

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

Calvin Lin Staff - 4 years, 4 months ago

2 is the only even prime number

Michael Mendrin - 4 years, 4 months ago

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

Michael Mendrin - 4 years, 4 months ago

@Michael Mendrin 4 is the maximum number of dimensions of a real division algebra

Michael Mendrin - 4 years, 4 months ago

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

Garrett Clarke - 4 years, 4 months ago

@Garrett Clarke 6 is the first perfect number!

Garrett Clarke - 4 years, 4 months ago

@Garrett Clarke 7 is the number of all possible Frieze groups

Michael Mendrin - 4 years, 4 months ago

@Michael Mendrin 8 is the first cube of a prime!

Garrett Clarke - 4 years, 4 months ago

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

Michael Mendrin - 4 years, 4 months ago

@Michael Mendrin 10 is the base that humans use to count.

Calvin Lin Staff - 4 years, 4 months ago

@Calvin Lin 11 is the smallest integer palindrome

Michael Mendrin - 4 years, 4 months ago

@Michael Mendrin 12 is the number of possible 3x3 latin squares!

Garrett Clarke - 4 years, 4 months ago

@Garrett Clarke 13 is the number of all possible Archimedian solids

Michael Mendrin - 4 years, 4 months ago

@Michael Mendrin 14 is the sum of the first three square numbers!

Garrett Clarke - 4 years, 4 months ago

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

Michael Mendrin - 4 years, 4 months ago

@Michael Mendrin 16 is the first non-trivial Erdős-Woods number!

Garrett Clarke - 4 years, 4 months ago

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

Michael Mendrin - 4 years, 4 months ago

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

Garrett Clarke - 4 years, 4 months ago

@Garrett Clarke 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!

Michael Mendrin - 4 years, 4 months ago

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

Brian Charlesworth - 4 years, 4 months ago

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

Garrett Clarke - 4 years, 4 months ago

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

Brian Charlesworth - 4 years, 4 months ago

@Brian Charlesworth 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)

Satyajit Mohanty - 4 years, 4 months ago

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

Brian Charlesworth - 4 years, 4 months ago

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

Satyajit Mohanty - 4 years, 4 months ago

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

shivamani patil - 4 years, 4 months ago

@Shivamani Patil 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.

Satyajit Mohanty - 4 years, 4 months ago

@Satyajit Mohanty 28 is the sum of the totient function for the first nine integers

shivamani patil - 4 years, 4 months ago

@Shivamani Patil 29 is the sum of three consecutive squares!

Garrett Clarke - 4 years, 4 months ago

@Garrett Clarke 30 is the smallest sphenic number.

shivamani patil - 4 years, 4 months ago

@Shivamani Patil 31 is a centered triangular number, a centered pentagonal number and centered decagonal number.

Satyajit Mohanty - 4 years, 4 months ago

@Satyajit Mohanty 32 is the ninth happy number.

shivamani patil - 4 years, 4 months ago

@Shivamani Patil 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.

Satyajit Mohanty - 4 years, 4 months ago

@Satyajit Mohanty 34 is the ninth Fibonacci number

shivamani patil - 4 years, 4 months ago

@Shivamani Patil 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.

Satyajit Mohanty - 4 years, 4 months ago

@Satyajit Mohanty 36 is the square of the first perfect number.

Adarsh Kumar - 4 years, 4 months ago

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

Satyajit Mohanty - 4 years, 4 months ago

@Satyajit Mohanty 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.

Pranshu Gaba - 4 years, 4 months ago

@Pranshu Gaba 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}.

Satyajit Mohanty - 4 years, 4 months ago

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

Brian Charlesworth - 4 years, 4 months ago

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

Pranshu Gaba - 4 years, 4 months ago

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

Brian Charlesworth - 4 years, 4 months ago

@Brian Charlesworth 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.

Satyajit Mohanty - 4 years, 4 months ago

@Brian Charlesworth 43 is the first centered heptagonal prime.

Pranshu Gaba - 4 years, 4 months ago

@Pranshu Gaba 44 is the number of derangements of 5 distinct objects.

Brian Charlesworth - 4 years, 4 months ago

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

Pranshu Gaba - 4 years, 4 months ago

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

Brian Charlesworth - 4 years, 4 months ago

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

9999999999999999999999999999999999999999999999212765957446808510638297872340425531914893617=47\large{\dfrac{9999999999999999999999999999999999999999999999}{212765957446808510638297872340425531914893617} = 47} .

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

Satyajit Mohanty - 4 years, 4 months ago

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

Maggie Miller - 4 years, 4 months ago

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

Satvik Golechha - 4 years, 4 months ago

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

Maggie Miller - 4 years, 4 months ago

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

Brian Charlesworth - 4 years, 4 months ago

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

Maggie Miller - 4 years, 4 months ago

@Maggie Miller The sum of the first 53 primes is divisible by 53.

Pranshu Gaba - 4 years, 4 months ago

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

Nihar Mahajan - 4 years, 4 months ago

@Nihar Mahajan 55 is the largest Fibonacci number that is triangular.

Maggie Miller - 4 years, 4 months ago

@Maggie Miller 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).

Satyajit Mohanty - 4 years, 4 months ago

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

Zi Song Yeoh - 4 years, 4 months ago

@Zi Song Yeoh 58 is the sum of first 7 prime numbers.

Nihar Mahajan - 4 years, 4 months ago

@Nihar Mahajan There are 59 stellations of the icosahedron.

Nihar Mahajan - 4 years, 4 months ago

@Nihar Mahajan There are 60 one-sided hexominoes.

Pranshu Gaba - 4 years, 4 months ago

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

Brian Charlesworth - 4 years, 4 months ago

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

Pranshu Gaba - 4 years, 4 months ago

@Pranshu Gaba 63 is the smallest positive integer NN such that Nk,1k9\frac{N}{k}, 1 \le k \le 9 has no repeating decimals.

Brian Charlesworth - 4 years, 4 months ago

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

Pranshu Gaba - 4 years, 4 months ago

@Pranshu Gaba 65 is equal to 15+24+33+42+51.1^{5} + 2^{4} + 3^{3} + 4^{2} + 5^{1}.

Brian Charlesworth - 4 years, 4 months ago

@Brian Charlesworth 66 is the first sphenic triangular number.

Pranshu Gaba - 4 years, 4 months ago

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

Satyajit Mohanty - 4 years, 4 months ago

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

Pranshu Gaba - 4 years, 4 months ago

@Pranshu Gaba 69 is an integer such that its square and cube concatenated contains each of 00 through 99 exactly once.

Brian Charlesworth - 4 years, 4 months ago

@Brian Charlesworth 70 is the smallest weird number.

Pranshu Gaba - 4 years, 4 months ago

@Pranshu Gaba 71 is conjectured to be the largest number whose square is a factorial plus one, (712=7!+171^{2} = 7! + 1).

Brian Charlesworth - 4 years, 4 months ago

@Brian Charlesworth 72: Smallest number whose fifth power is the sum of five smaller fifth powers (195+435+465+475+675=725)(19^5 + 43^5 + 46^5 + 47^5 + 67^5 = 72^5).

Pranshu Gaba - 4 years, 4 months ago

@Pranshu Gaba 73 is the smallest positive integer nn such that each of n1,nn - 1, n and n+1n + 1 are expressible as sums of two squares.

Brian Charlesworth - 4 years, 4 months ago

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

Pranshu Gaba - 4 years, 4 months ago

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

Brian Charlesworth - 4 years, 4 months ago

@Brian Charlesworth 76 is an automorphic number because 752{75}^{2}'s last 2 digits is 75.

Arulx Z - 4 years, 4 months ago

@Arulx Z 77 is the sum of first 8 prime numbers.

Pranshu Gaba - 4 years, 4 months ago

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

Arulx Z - 4 years, 4 months ago

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

Pranshu Gaba - 4 years, 4 months ago

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

Adarsh Kumar - 4 years, 4 months ago

@Adarsh Kumar 81 is the square of the sum of its digits.

Arulx Z - 4 years, 4 months ago

@Arulx Z 82:a happy number

Adarsh Kumar - 4 years, 4 months ago

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

Arulx Z - 4 years, 4 months ago

@Arulx Z A hepteract has 84 penteract 5-faces.

Pranshu Gaba - 4 years, 4 months ago

@Pranshu Gaba 85 is the largest nn for which \sum _{ x=1 }^{ n }{ { x }^{ 2 } } =\sum _{ y=1 }^{ m }{ y } for some mm.

Arulx Z - 4 years, 4 months ago

@Arulx Z 2862^{86} is conjectured to be the highest power of 2 without a 0 in its decimal expansion.

Pranshu Gaba - 4 years, 4 months ago

@Pranshu Gaba 87 is the sum of the squares of the first 4 primes.

Arulx Z - 4 years, 4 months ago

@Arulx Z 88 is conjectured to be the only number whose square has no "isolated" digits. (882=7744.88^{2} = 7744.)

Brian Charlesworth - 4 years, 4 months ago

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

Satyajit Mohanty - 4 years, 4 months ago

@Satyajit Mohanty 90 is the sum of its digits plus the squares of its digits.

Arulx Z - 4 years, 4 months ago

@Arulx Z 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=63+(5)3=43+3391 = 6^3+(-5)^3 = 4^3+3^3

91 is the smallest positive integer expressible as a sum of six distinct squares: 91=12+22+32+42+52+6291 = 1^2+2^2+3^2+4^2+5^2+6^2

91 is the smallest pseudoprime satisfying the congruence 3n3(modn)3^n \equiv 3 \pmod{n}.

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

Satyajit Mohanty - 4 years, 4 months ago

@Satyajit Mohanty There are 92 naturally occurring elements on earth.

Nihar Mahajan - 4 years, 4 months ago

@Nihar Mahajan 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.

Satyajit Mohanty - 4 years, 4 months ago

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

Nihar Mahajan - 4 years, 4 months ago

@Nihar Mahajan 95 is a Thabit number.

Shenal Kotuwewatta - 4 years, 4 months ago

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

Shenal Kotuwewatta - 4 years, 4 months ago

@Shenal Kotuwewatta 97 is the largest two-digit prime.

Shenal Kotuwewatta - 4 years, 4 months ago

@Shenal Kotuwewatta 98 is not in the range of ϕ\phi.

Shenal Kotuwewatta - 4 years, 4 months ago

@Shenal Kotuwewatta 99 is the sum of the cubes of three consecutive integers:

Shenal Kotuwewatta - 4 years, 4 months ago

@Shenal Kotuwewatta 100 is the smallest three-digit number.

Shenal Kotuwewatta - 4 years, 4 months ago

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

Shenal Kotuwewatta - 4 years, 4 months ago

@Shenal Kotuwewatta 102 contains each of the first three digits exactly once.

Alex Li - 4 years, 4 months ago

@Alex Li 103 is the smallest cyclops prime with distinct digits

Michael Mendrin - 4 years, 4 months ago

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

Michael Mendrin - 4 years, 4 months ago

@Michael Mendrin 105 is the smallest dodecagonal number that is also a double factorial

Michael Mendrin - 4 years, 4 months ago

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

Michael Mendrin - 4 years, 4 months ago

@Michael Mendrin 106 is the number of distinct trees with ten vertices.

Alex Li - 4 years, 4 months ago

@Alex Li 107 is the smallest three digit Mersenne prime exponent

Michael Mendrin - 4 years, 4 months ago

@Michael Mendrin 108 is the interior angle of the regular pentagon

Michael Mendrin - 4 years, 4 months ago

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

Michael Mendrin - 4 years, 4 months ago

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

Michael Mendrin - 4 years, 4 months ago

@Michael Mendrin 111 is the smallest repunit semiprime

Michael Mendrin - 4 years, 4 months ago

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

Michael Mendrin - 4 years, 4 months ago

@Michael Mendrin 113 is the smallest three digit permutable prime

Michael Mendrin - 4 years, 4 months ago

@Michael Mendrin 114 is the sum of the first four hyperfactorials

Michael Mendrin - 4 years, 4 months ago

@Michael Mendrin 115 is the smallest odd heptagonal pyramidal semiprime

Michael Mendrin - 4 years, 4 months ago

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

Brian Charlesworth - 4 years, 4 months ago

@Brian Charlesworth 117 is the smallest pentagonal number divisible by an odd square

Michael Mendrin - 4 years, 4 months ago

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

Michael Mendrin - 4 years, 4 months ago

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

Alex Li - 4 years, 4 months ago

@Alex Li 120 is 5!5!

Alex Li - 4 years, 4 months ago

@Alex Li 121 is the smallest nontrivial palindromic square.

Alex Li - 4 years, 4 months ago

@Arulx Z 76 raise to power any positive integer end with 76

Ravneet Singh - 4 years, 4 months ago

@Garrett Clarke 30 is the sum of first four squares.

Karthik Venkata - 4 years, 4 months ago

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

Satyajit Mohanty - 4 years, 4 months ago
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