Wikipedia:1729

Just about anyone even casually acquainted with number theory knows the anecdote about the mathematicians Hardy and Ramanujan talking about the seeming uninterestingness of the number 1729.

In the context of Wikipedia, Wikipedia:WikiProject Numbers asks that three interesting properties of a number be gathered before even considering creating an article on that number.

Sometimes there is agreement that a given mathematical property is interesting (e.g., that 1729 is the sum of two cubes in two different ways), so there is no problem. Other times there is disagreement, and some way of measuring the interestingness of a property in relation to a given number is necessary.

Hopefully the following questionnaire will prove useful in those situations, providing help in evaluating how interesting an integer's mathematical property is. Do note that the purpose of this questionnaire is to help determine if a mathematical property is interesting enough to create an article on the given number. It might be acceptable for an article on a number to mention properties that were not deemed interesting enough to justify the article in the first place, as long as the properties that were interesting enough are also mentioned.

The questionnaire

Number N has the mathematical property that the Boolean function f(N) = True.

1. How many n < 10⁷ do NOT have this property in common with Number N? If it's too computationally intensive to calculate, a heuristic estimate is acceptable, or even a rough guesstimate. This total is the initial number of points given to the mathematical property of the integer.

2. Has a professional mathematician written a peer-reviewed paper or book about this property that specifically mentions Number N?

YES. What is the Erdős number Ő of the mathematician? (If Erdős himself, let Ő = 1 here to avoid division by 0 at this step). Divide the points from Question 1 by Ő and round off if needed. Alternatively, because earlier mathematicians (like Leonard Euler) do not have Erdős numbers, assign a mathematician with a top-priority article Ő = 1, high-priority Ő = 3, medium priority Ő = 5, and low/unassessed priority Ő = 10. If a mathematician is notable enough for inclusion on Wikipedia but does not have a known Erdős number, assume Ő = 10.

NO. Deduct 10⁷ points.

3. In the list of numbers with the property, sorted in ascending order, at what position k does Number N occur? Deduct (k − 1) from Question 2 points.

4. Might f(N) = False in a different base b?

NO. Skip ahead to Question 5.

YES. For bases 1 < b < 17, compute f(N). For each True award b points. For each False deduct bN points.

5. Does the sequence of numbers with f(N) = True in Sloane's OEIS specifically list Number N in its Sequence or Signed field?

YES. Award the A-number of the sequence as points.

NO. Skip to question 7.

6. What keywords does the sequence have in its Keywords field?

core. Subtract the sequence's A-number from the A-number of the most recently added sequence. Award that difference as points.

nice. Award the A-number of the sequence as points.

hard. Award the A-number of the sequence as points again.

more. Award the A-number of the sequence as points again.

base. Make sure you did not skip Question 4.

less. Deduct the sequence's A-number as points.

Any others. Award a point each.

7. How many points are there?

points > 0. The property in relation to the number is interesting.

points = 0. It's your call.

points < 0. The property in relation to the number is NOT interesting.

Examples

1729

For the sake of example, suppose that there isn't an article on the number 1729. Sally has jotted down a few properties of the number, namely:

• 1729 is odd.
  • 1. Starting with 5 × 10⁶ points.
    2. Mathematicians have certainly written papers about parity, but Sally doesn't care to look for one that specifically mentions 1729. So 10⁷ points are deducted, leaving -5 × 10⁶ points.
    3. In the list of odd numbers, 1729 occurs at position 865, so now there are -5000864 points.
    4. 1729 is odd regardless of base, so the question is skipped.
    5. The largest odd integer in the sequence field of Sloane's Tɛmplet:OEIS is 131.
    6. Question skipped.
    7. There are -5000864 points, meaning that it's not interesting that 1729 is odd.

• 1729 is a Carmichael number.
  • 1. 512461 is the 33rd Carmichael number, so Sally guesstimates that there are about 65 Carmichael numbers below 10⁷. So Sally starts with 9999935 points.
    2. Wacław Sierpiński wrote a paper entitled "A Selection of Problems in the Theory of Numbers." The index tells Sally that the paper deals with Carmichael numbers on page 51. Sierpiński has Erdős number 2, so multiply the starting points by 1/2, so now there are 4999968 points.
    3. In the list of Carmichael numbers, 1729 is third. Now there are 4999966 points.
    4. Question skipped.
    5. 1729 indeed appears in Sloane's Tɛmplet:OEIS2C. Award 2997 points, bringing total up to 5002963.
    6. A002997 has the keyword nice, so award another 2997 points. It also has the keywords nonn and easy, so award 2 points.
    7. There are 5005962 points, meaning that it is interesting that 1729 is a Carmichael number.
       
• 1729 is a Harshad number.
  • 1. There are 11872 Harshad numbers below 10⁵, so Sally guesstimates that there are 1187200 below 10⁷. So begin with 8812800 points.
    2. Sally assumes that although mathematicians have written papers on Harshad numbers and on 1729, probably none has written a paper on the fact that 1729 is a Harshad number. So we're down to -1187200 points.
    3. 1729 is the 364th Harshad number, bringing us further down to -1187563 points.
    4. 1729 is also a Harshad number in bases 4, 5, 7, 8, 13, and 16. That brings us up to -1187510. But it's not Harshad in bases 2, 3, 6, 9, 11, 12, 14, or 15, bringing the points down to -1291250.
    5. The largest Harshad number in the sequence field of Sloane's Tɛmplet:OEIS2C is 204.
    6. Question skipped.
    7. There are -1291250 points, meaning that it's not interesting that 1729 is a Harshad number.

• 1729 is a taxicab number, it can be expressed as a sum of two cubes in two different ways.
  • 1. There are ten such numbers below 10⁵, so Sally guesstimates that there are 1000 of them below 10⁷. So begin with 9999000 points.
    2. G. H. Hardy wrote about this property of 1729 in his book about Ramanujan's lectures. Hardy has Erdős number Ő = 2. Now we're at 4999500 points.
    3. 1729 is the very first number with this property, so this doesn't even make a dent to the points.
    4. Question skipped.
    5. Sloane's Tɛmplet:OEIS2C has 1729 in its Sequence field, bringing the points up to 5000735.
    6. The Keyword field has the keyword "nice," so award another 1235 points, plus 1 point for the keyword "nonn."
    7. There are 5001971 points, meaning that it is interesting that 1729 can be expressed as a sum of two cubes in two different ways.

• 1729 is a Zeisel number.
  • 1. There are 24 Zeisel numbers less than a million, so Sally guesstimates there are 240 less than ten million. So begin with 9999760 points.
    2. The only printed reference Sally can find is in Eric W. Weisstein's CRC Concise Encyclopedia of Mathematics. Sally doesn't know Weisstein's Erdős number, but 10 is probably too high. So dividing the points by 10, there are 999976 points.
    3. 1729 is the third Zeisel number, so now there are 999974 points.
    4. Question skipped.
    5. Sloane's Tɛmplet:OEIS2C has 1729 in its Sequence field, bringing the points up to 1050989.
    6. The only keyword is "nonn."
    7. There are 1050990 points, meaning that it is interesting that 1729 is a Zeisel number.

Therefore, Sally has gathered three interesting properties of 1729. She might be ready to create an article on 1729, though she reads WP:NUM for further advice.

170141183460469231731687303715884105727

Dick wants to write a Wikipedia article on the double Mersenne prime 170141183460469231731687303715884105727.

• 170141183460469231731687303715884105727 is a double Mersenne prime
  • 1. There are only 2 among the first 10⁷ integers, so Dick starts with 9999998 points.
    2. Pomerance and Crandall specifically mention this number in their book Prime numbers: a computational perspective. Pomerance has Erdős number 1, so there are still 9999998 points.
    3. 170141183460469231731687303715884105727 is the fourth double Mersenne prime, so we're down to 9999995 points.
    4. Since we're inquiring on numbers of the form $2^{2^{p}-1}-1$ and not binary repunits, this question doesn't apply.
    5. This number appears in Sloane's Tɛmplet:OEIS2C. 10077581 points.
    6. The only keyword for Sloane's Tɛmplet:OEIS2C is nonn. 10077582 points.
    7. There are 10077582 points, meaning that it is interesting that 170141183460469231731687303715884105727 is a double Mersenne prime.

So there's one interesting property for 170141183460469231731687303715884105727. But Dick needs two more before he can justify writing a Wikipedia article on this number.

A hypothetical second odd perfect number

Suppose Tom discovers two odd perfect numbers OP₁ and OP₂. There's no doubt that the first odd perfect number deserves its own article. But does the second?

• OP₂ is an odd perfect number.
  • 1. OP₂ would have to be at least 10³⁰⁰, so it's a safe bet to start with 10⁷ points.
    2. Mathematicians had some idea that OP₂ has as many factors as it does, but they couldn't possibly know exactly, otherwise they would have discovered it, not Tom. So 10⁷ points are deducted, leaving none.
    3. Since OP₂ is the second odd perfect number, we're now down to -1 points.
    4. Question skipped.
    5. OP₂ doesn't appear in Sloane's OEIS at all.
    6. Question skipped.
    7. There are -1 points, meaning that it's not interesting that OP₂ is an odd perfect number.

This is not the end of the story, however. If the premise had been that OP₂ is odd, the questionnaire would've ended up with at least -10³⁰⁰ points. So an answer of -1 points is not as conclusive as an answer of -10³⁰⁰ points.

Since OP₂ would be a major discovery, it would be inevitable that mathematicians would start studying this number, even if many of them quickly dismissed Tom out of hand as an amateur. They might even find other interesting properties of OP₂ besides its being an odd perfect number.

But if OP₂ has no other interesting properties, there's no reason to give it its own article.

1023458967

Harry wants to create an article on the pandigital number 1023458967. The only property of the number that he knows about is that it's a pandigital number.

• 1023458967 is a pandigital number.
  • 1. Start with 10⁷ points.
    2. Harry finds an entry on pandigital numbers in Eric W. Weisstein's CRC Concise Encyclopedia of Mathematics. He ignores the fact that 1023458967 is not explicitly mentioned, and proceeds to ask Sally what Weisstein's Erdős number is. She says she guessed 10. This brings the points down to 10⁶.
    3. 1023458967 is the 17th pandigital number, so now we're down to 999984 points.
    4. 1023458967 is pandigital in bases 2, 3, 4, 5, 6 and of course 10, bringing the points up to 1000014. But it's not pandigital in bases 7, 8, 9 nor 11 to 16, bringing the points down to -107462191521.
    5. Sloane's Tɛmplet:OEIS2C has 1023458967 in its Sequence field. This brings the points up to -107462141243.
    6. The Keywords field reads "nonn, base, fini." 2 points for "nonn" and "fini" together. Harry doublechecks that he went through Question 4, and although the questionnaire says nothing about awarding points for the "base" keyword, Harry decides to award the square of the sequence's A-number as points anyway.
    7. Even that is not enough to bring the points out of the negative side, and with -104934263957 points it is inescapable to conclude that it's not that interesting that 1023458967 is a pandigital number.

103

This is not to say that base-dependent mathematical properties are always uninteresting. Suppose Harry wants to write an article on 103, and he decides to zoom in on the fact that 103 is not a palindromic number.

• 103 is not palindromic.
  • 1. There are 1098 palindromic numbers less than ten thousand, so Harry guesstimates that there are 109800 palindromics less than ten million. Thus 109800 are the starting points.
    2. Harry uses the entry on palindromic numbers in Weisstein's Encyclopedia even though it doesn't actually mention 103. 10980 points.
    3. 103 is the 84th non-palindromic number. Now at 10897 points.
    4. As it happens, 103 is not palindromic in any base from 2 to 16. (In fact, it's not palindromic until base 102). This brings the points up to 24802.
    5. Sloane's Tɛmplet:OEIS2C has 103 in its Sequence field. This brings the points up to 54544.
    6. The Keywords field reads "nonn, base, easy, nice."
    7. There are 84289 points, meaning that it is actually interesting that 103 is not palindromic.

What Harry has unwittingly stumbled on is that 103 is a strictly non-palindromic number.

See also

• Interesting number paradox