π: MATH Pages of Jonathan Vos Post



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"This sentence contains ten words, eighteen syllables, and sixty-four letters."

[Jonathan Vos Post, Scientific American, reprinted in "Metamagical Themas: Questing for the Essence of Mind and Pattern", by Douglas R. Hofstadter, paperback reprint March 1996, pp.26-27] Jonathan Vos Post is a Professor of Mathematics at Woodbury University in Burbank, California. His first degree in Mathematics was from Caltech in 1973. He is also, or has been also, a Professor of Astronomy at Cypress College in Orange County, California; Professor of Computer Science at California State University, Los Angeles; and Professor of English Composition at Pasadena City College. He is a widely published author of Science Fiction, Science, Poetry, Math, Drama, and other fields. In his so-called spare time, he wins elections for local political offices and produces operas, as Secretary of Euterpe Opera Theatre. His Erdos Number is 5. For more on Erdos Numbers and related topic, visit the blog at:

magicdragon2: Erdos & Asimov

Hotlinks to Math Pages of Jonathan Vos Post

Table of Figurate Numbers, Sorted, Through 10,000

I created this table as a useful and fun reference list while working on several of the Number Theory papers listed below. It goes far beyond 10,000 for many kinds of 2-D and 3-D Figurate Numbers.

Table of Polytope Numbers, Sorted, Through 1,000,000

I created this 400 Kilobyte table of 4-D, 5-D, 6-D, 7-D, 8-D, 9-D, and 10-D Polytope Numbers as a useful and fun reference list while working on several of the Number Theory papers listed below. It goes far beyond 1,000,000 for some sequences.

Four Nines Puzzle

"The Weekly Dispatch" of 4 February 1900, which ran a puzzle column by Dudeney, introduced a problem which is still provoking interest today. That problem was the Four Nines Puzzle, based on the even older Four Fours Puzzle, which is also discussed on this web page. Jonathan's page hotlinked here includes a complete list of equations representing, with four nines, every integer up to 314, and many beyond that. There is some deep theory towards the bottom of the page.

Four π Puzzle

Similar to the Four Nines Puzzle, and Four Fours Puzzle, mentioned above, What numbers can be made with four copies of the number "pi", or π? Jonathan's page hotlinked here includes a complete list of equations representing, with four "pi", every integer up to 1,000, and many beyond that. It is a useful teaching tool for geometry or algebra students to ask: "How can we construct the smaller whole numbers, under 100 for instance, using only all four copies of the number π, parentheses, and the arithmetic operators

"+","-","x","/"?

We also allow the use of exponentiation, radicals (especially the square root "sqrt"), factorial "!", and the floor function "|_ N _|" and ceiling function "|- N -|".

MATHEMATICS: Fantasy and Science Fiction about Mathematics

Warning: the above-linked page is over 300 Kilobytes long, and may load slowly. That's because it has not only the listed topic, but an original encyclopedia of many other subgenres of Science Fiction and Fantasy.

COMING SOON: Math papers I've written and submitted whose reprint/preprints will be posted here or hotlinked

Inventory Status Summary: 43 Math papers written in past year; 9 presented and accepted in proceedings of International Conferences; 1 published as 3 peer-reviewd pages in prestigious web encyclopedias; 4 still in editorial hands at Mathematics Magazine [#6,7,19,30]; 3 rejected by Mathematics Magazine, being resubmitted elsewhere; 1 still in editorial hands at Fibonacci Quarterly [#31]; 1 rejected by Fibonacci Quarterly, being resubmitted elsewhere; 2 rejected by American Mathematical Monthly, being resubmitted elsewhere; 19 still being completed/polished for first submission. Coming Soon #1:

Iterated Triangular Numbers

[submitted to Mathematics Magazine, January 2004; manuscript number 03-1065 per email 8 Jan 2004; rejected; resubmitted to : _______________] Coming Soon #2:

Iterated Polygonal Numbers

[submitted to Mathematics Magazine, January 2004; manuscript number 03-1074 per email 28 Jan 2004; rejected; resubmitted to : _______________] Coming Soon #3:

Triangular Carmichael Numbers: The First 22 Identified

[submitted to American Mathematical Monthly, 20 February 2004; rejected; resubmitted to _____________________] Coming Soon #4:

Triangular Dodecagonal Numbers

[submitted to Mathematics Magazine, 23 February 2004; manuscript number 04-1102 per email 2 Mar 2004; rejected; resubmitted to : _______________] Coming Soon #5:

Dodecagonal Squares from Fibonacci Numbers

[submitted to Fibonacci Journal, 23 February 2004; rejected; resubmitted to : _______________] Coming Soon #6:

When Dodecagonal Numbers are Pentagonal Numbers

[research complete 19 March 2004, submitted to Mathematics Magazine, 24 March 2004; manuscript number 04-1125 per email 29 Mar 2004]] Coming Soon #7:

When 24-gonal Numbers are Perfect Squares

[research complete 31 March 2004, submitted to Mathematics Magazine, 16 May 2004; manuscript no. 04-1177, per email 25 May 2004] Coming Soon #8:

Dodecahedron Numbers which are Perfect Squares

[written 18 March 2004; submitted to _____________, __ March 2004] Coming Soon #9:

When Icosahedron Numbers are Perfect Squares

[written 19 March 2004; submitted to _____________, __ March 2004] Coming Soon #10:

When Pentatope Numbers are Pentagonal Numbers

[submitted to American Mathematical Monthly, 8 March 2004; rejected; resubmitted to _____________________] Coming Soon #11:

Pentatope Numbers, Simplex Numbers, and Polynomial Factoring

[main theorems proved by __ May 2004; submitted to ________, __ May 2004] Coming Soon #12:

When Pentatope Numbers are Polygonal Pyramidal Numbers

[main theorem proved 1 May 2004; submitted to _________, __ May 2004] [Related to #14] Coming Soon #13:

When Polygonal Pyramidal Numbers are also Polygonal Pyramidal Numbers of Different Rank

[10 main theorems proved by 10 May 2004; submitted to _________, __ May 2004] Coming Soon #14:

When Hyperoctahedron Numbers are Octagonal Numbers

[submitted to American Mathematical Monthly, 9 March 2004; rejected; resubmitted to _____________________] [Related to #12] Coming Soon #15:

Reversion of Cross-Polytope Numbers, and when d-Dimensional Hyperoctahedron Numbers are Perfect Squares

[written 17 March 2004; submitted to ________________, __ March 2004] Coming Soon #16:

When Hyperdodecahedron Numbers are Perfect Squares

[written __ March 2004; submitted to _____________, __ March 2004] Coming Soon #17:

When Hypericosahedron Numbers are Perfect Squares

[written __ March 2004; submitted to _____________, __ March 2004] Coming Soon #18:

When Hyperdiamond Numbers are Perfect Squares

[written __ March 2004; submitted to _____________, __ March 2004] Coming Soon #19:

When Centered Polygonal Numbers are Perfect Squares

[submitted to Mathematics Magazine, 4 May 2004, manuscript no. 04-1165, per email 6 May 2004] Coming Soon #20:

Collisions Between Classes of Integer Sequences

(a work in progress, 25+ pages long as of 3 May 2004, spun-off from "When Pentatope Numbers are Polygonal Pyramidal Numbers" and puts in context most of the above-titled papers) Coming Soon #21:

Imaginary Mass, Momentum, and Acceleration: Physical or Nonphysical?

[Proceedings of the Fifth International Conference on Complexity Science, 17-21 May 2004] Co-author #1 = Andrew Carmichael Post Co-author #2 = Professor Christine Carmichael, Woodbury University Coming Soon #22:

The Evolution of Controllability in Enzyme System Dynamics

[Proceedings of the Fifth International Conference on Complexity Science, 17-21 May 2004] Coming Soon #23:

Adaptation and Coevolution on an Emergent Global Competitive Landscape

[Proceedings of the Fifth International Conference on Complexity Science, 17-21 May 2004] Author #1 = Professor Philip V. Fellman, Southern New Hampshire University Author #2 = Professor Jonathan Vos Post, Woodbury University Author #3 = Roxana Wright, Southern New Hampshire University Coming Soon #24:

The Nash Equilibrium Revisited: Chaos and Complexity Hidden in Simplicity

[Proceedings of the Fifth International Conference on Complexity Science, 17-21 May 2004] Author #1 = Professor Philip V. Fellman, Southern New Hampshire University Author #2 = Professor Jonathan Vos Post, Woodbury University Coming Soon #25:

Competitive Intelligence: A Game Theoretic Approach

[Manuscript, 2003, in submission/rewrite process] Author #1 = Professor Philip V. Fellman, Southern New Hampshire University Author #2 = Professor Jonathan Vos Post, Woodbury University Coming Soon #26:

Network Externalities, Evolutionary Dynamics, and Firm Internationalization

[Proceedings, Northeast Regional Meeting of the Academy of International Business, 2003] Author #1 = Professor Philip V. Fellman, Southern New Hampshire University Author #2 = Professor Jonathan Vos Post, Woodbury University Coming Soon #27:

The Implications of Peter Lynds 'Time and Classical and Quantum Mechanics: Indeterminacy vs Discontinuity' for Mathematical Modeling"

[Proceedings, North American Association for Computation in the Social and Organizational Sciences, 2004] Author #1 = Professor Philip V. Fellman, Southern New Hampshire University Author #2 = Maurice Passman Author #3 = Professor Jonathan Vos Post, Woodbury University Author #4 = Professor Christine Carmichael, Woodbury University Author #5 = Andrew Carmichael Post, California State University Los Angeles Coming Soon #28:

The Nash Equilibrium: Polytope Decision Spaces and Non-linear and Quantum Computational Architectures

[Proceedings of the Fifth International Conference on Complexity Science, 17-21 May 2004] [title corrected from earlier draft of webpage] Author #1 = Professor Philip V. Fellman, Southern New Hampshire University Author #2 = Professor Jonathan Vos Post, Woodbury University Coming Soon #29:

'Time and Classical and Quantum Mechanics' and the Arrow of Time

[Proceedings, North American Association for Computation in the Social and Organizational Sciences, 2004] Author #1 = Professor Philip V. Fellman, Southern New Hampshire University Author #2 = Professor Jonathan Vos Post, Woodbury University Coming Soon #30:

Semiprime Figurate Numbers

[submitted to Mathematics Magazine, 19 July 2004] Assigned manuscript #04-1218 per email to me of 23 July 2004 Coming Soon #31:

Fibonacci Semiprimes

[submitted to Fibonacci Quarterly, 4 Aug 2004] Coming Soon #32:

Semiprime Lucas Numbers

[submitted to Fibonacci Quarterly, XX Aug 2004] Coming Soon #33:

Semiprime Triangular Numbers

Author #1 = Professor Jonathan Vos Post, Woodbury University Author #2 = Dr. Geoffrey A. Landis, John H. Glenn NASA Center [current draft 4 of 29 July 2004] Coming Soon #34:

Concatenated Semiprime Numbers

[current draft of 5 August 2004] Coming Soon #35:

Primes, Semiprime, and Factorizations of N-dimensional Centered Tetrahedral and Centered Cube Numbers

[current 49-page draft 3.0 of 11 August 2004] Linked Now: #36:

Emirpimes

Eric W. Weisstein and Jonathan Vos Post, "Emirpimes." From MathWorld -- a Wolfram Web Resource

Emirpimes

which in turn hotlinks to N.J.A. Sloan's

"The On-Line Encyclopedia of Integer Sequences"

, Sequences A097393 and A097394 I have much more information on Emirpimes, which will eventually be written for journal publication [manuscript #37] Coming Soon: #37:

Semiprimes and Emirpimes with the "10201" Digit Pattern

[draft 1.0: research, writing, emailing done 19 Aug 2004] Coming Soon: #38:

Semiprime Smith Numbers and Some Conjectures

[draft 1.0: research, writing, emailing done 24 Aug 2004] Coming Soon: #39:

Semiprime Keith Numbers

[draft 1.0: research, writing, emailing done 25 Aug 2004] Coming Soon: #40:

Semiprime Kynea Numbers

[draft 1.0: research, writing, emailing done 26 Aug 2004; more calculations overnight for draft 2.0] Coming Soon: #41:

Iterated Sums of Squares of Prime Factors

[draft 1.0: research, writing, emailing done 26 Aug 2004 Coming Soon: #42:

Asimov Numbers: Notes Towards a Small World Graph of Science Fiction Authors

[draft 1.0: research, writing, emailing done 28 May-???? 2004] [to be submitted to Feb 2005 Conference at Redondo Beach on Social Network Theory Coming Soon #43:

My Erdos Number, and how I connect to Erdos through Three Nobel laureates in Physics.

My Erdos Number is 5, because I coauthored with Richard Feynman, who coauthored with Murray Gell-Mann, who coauthored with Sheldon Lee Glashow, who coauthored with Daniel J. Kleitman, who coauthored with Paul Erdos. Specifically: (5) Jonathan Vos Post and Richard P. Feynman [Nobel Prize, Physics, 1965]: "Footnote to Feynman", Jonathan V. Post and Richard Feynman, [Engineering & Science, Caltech, Pasadena, CA, Vol.XLVI, No.5, p.28, ISSN: 0013-7812, May 1983; reprinted in Songs from Unsung Worlds, ed. Bonnie Bilyeu Gordon, intro by Alan Lightman (award winning author of Einstein's Dreams), Birkhauser Boston/AAAS, hardcover ISBN: 0-8176-3296-4, paperback ISBN: 3-7643-3296-4, 1985 (4) Richard P. Feynman with Murray Gell-Mann [Nobel Prize, Physics, 1969]: R. P. Feynman, M. Gell-Mann, & G. Zweig, "Group U(6)xU(6) generated by current components", Phys. Rev. Lett. 13(1964)678-680 (3) Murray Gell-Mann with Sheldon Lee Glashow [Nobel Prize, Physics, 1965?]: Sheldon L. Glashow & Murray Gell-Mann, "Gauge theories of vector particles," Ann. Physics 15(1961)437-460 (2) Sheldon Lee Glashow with Daniel J. Kleitman S. L. Glashow & D. J. Kleitman, "Baryon resonances in W3 symmetry", Phys. Lett. 11(1964)84-86 [Note: because of the above, Glashow shares with Albert Einstein the distinction of being, up to now, the only Nobel-winning physicists with Erdos number = 1 or 2] [Note: combinatorist Daniel J. Kleitman is also Glashow's brother-in-law] (1) Daniel J. Kleitman with Paul Erdos: 7 papers, earliest being 1965 P. Erdos, D. J. Kleitman & B. L. Rothschild, "Asymptotic enumeration of Kn-free graphs", in Intern. Colloq. Combin, Theo., Atti dei Convegni Lincei 17, Rom. [Note: I might also comment that both Gell-Mann and Erdos were among the roughly 1,500 scientists who signed the "World Scientists' Warning to Humanity", 18 November 1992]

Teaching Resume of Jonathan Vos Post

My Teachers' Teachers' Teachers

Some of the most famnous mathematicians in history were Jonathan's teachers of teachers of teachers... See this intellectual geneology. Coming Soon: Annotated Partial List of 65 Mathematics and Computer Science Publications and Presentations in my credits.

Hotlinks to Other Cool Math Pages

Periodic Table of the Mathematicians

What's Special About This Number

Delightful coloful table of almost all integers up to 10,000 with interesting facts about each one. I have contributed about 30 of these, which Professor Erich Freidman quite reasonably does not attribute (as he is interested in facts only on this page). But for the record, the 110 ones that I invented (or rediscovered in this context) and submitted to Dr. Friedman include:

12:

a Pentagonal, Dodecagonal, Hendecagonal Pyramidal Number

33:

the 33rd Triangular Number is also a Dodecagonal Carmichael Number

55:

the square root of a Square Dodecagonal Number

64:

a Square Dodecagonal Number

91:

the 91st Dodecagonal Number is also a Triangular Carmichael Number

105:

a Triangular Dodecagonal Number

137:

The reciprocal of a dimensionless constant in Atomic Physics is approximately 137.036. By dimensionless, I mean that this is just a number, unlike a distance in inches, which would be a different number of centimeters. Many people have wondered why this constant is what it is, and not larger or smaller; the question what is special about 137 has fascinated scientists. The constant of roughly 1/137.036 is called the Fine Structure Constant, because some of the things it measures are ceratin details of the spectrum of light emitted by glowing gases. Also: 137 = 10001 / 73.

152:

note that 1, 152, 11552 are the first 3 terms of Sloane's A035823, and that 1152 comes from repeating the 1st and 2nd digits of 152...

162:

See Center of Gravity With Integer Coordinates also comes up three times in 6-ary Lyndon words with given trace and subtrace

164:

10^2 + 8^2 = 20002 [base 3] = 1124 [base 5]. "E.164 addresses are used in networking ATMs. 164 comes up fairly early in Sloane's A000203.

167:

12 x 167 = 2004 [this year]. 167 comes up in the problem of integer square tilings, a tiling of a square with smaller squares of integer sides. For every n > 1 consider all square tilings of an n x n square, and define: f(n) = the largest possible size of the smallest square g(n) = the smallest number of squares h(n) = the smalles value of the largest multiplicity of any square needed. Them for prime P, f(167) = 21 (with the largest possible minimum square), and f(167) = 21 (with the fewest possible squares). See Erich Friedman's Mathmagic

169:

the 169th Dodecagonal Number is also a Square

173:

comes up twice in Erich Friedman's Mathmagic

179:

comes up in the theory of Odd Greedy Expanisons {to be done}

185:

"every integer exceeding 185 is representable [given in the proof of proposition 4] "d-complete sequences of integers", Erdos & Lewin, Mathematics of Computation, Vol.65, No.214, Apr 1996, pp.837-840.

196:

a Square Hendecagonal Number

204:

the square root of a square triangular number (see Sloan A001110)

210:

a Triangular Pentagonal Number

221:

1!^2 + 2!^2 + 3!^2

286:

the 286th Triangular Number is also a Dodecagonal Carmichael Number

288:

the 288th triangular number is also a square (see Sloan A001110). Also see "Heron Tetrahedra." Other numbers which come up in Heron simplices include these combinations of edge lengths: (203,195,148) and (888,875,533).

301:

the first Odd Roman Numeral, alphabetically. See Numbers in Recreational Linguistics

377:

the square root of a Square Dodecagonal Number

561:

the smallest Carmichael Number; also a Triangular Number, and Dodecagonal Number

588:

588^2 + 2353^2 = 5882353. "The Hermite-Serret Algorithm and 12^2 + 33^2 = 1233", Alf van der Poorten.

617:

1!^2 + 2!^2 + 3!^2 + 4!^2

619:

1! - 2! + 3! - 4! + 5! - 6!

651:

a Pentagonal Number

715:

a Pentagonal Hendecagonal Number

775:

the 775th Dodecagonal Number is also a Triangular Number

782:

a Pentagonal Number

809:

1!^5 + 2!^4 + 3!^3 + 4!^2 + 5!^1

852:

a Pentagonal Number

925:

a Pentagonal Number

1001:

a Pentagonal Number, the same reversed or upside-down

1005:

One Thousand Five is the smallest number whose English name uses the five vowels a, e, i, o, u, in any order. See Numbers in Recreational Linguistics

1025:

One Thousand Twenty-Five is the smallest number whose English name uses the six vowels a, e, i, o, u, y in any order. See Numbers in Recreational Linguistics

1080:

a Pentagonal Number, and the smallest number with 18 divisors

1084:

One Thousand Eighty-Four is the smallest number whose English name uses the five vowels a, e, i, o, u, in that order. See Numbers in Recreational Linguistics

1104:

One Thousand One Hundred Four is the smallest number whose English name is spelled with 25 letters. See Numbers in Recreational Linguistics

1117:

One Thousand One Hundred Seventeen is the smallest number whose English name is spelled with 30 letters. See Numbers in Recreational Linguistics

1156:

the 1156th Dodecagonal Number is also a Square

1162:

a Pentagonal Number

1189:

the square root of a Square Triangular number (see Sloan A001110)

1247:

a Pentagonal Number

1335:

a Pentagonal Number

1426:

a Pentagonal Number

1520:

a Pentagonal Number

1540:

a Triangular, Hexagonal, Decagonal, Hendecagonal Pyramidal Number

1617:

a Pentagonal Number

1681:

the 1681st triangular number is also a square (see Sloan A001110); and bot the initial pair of digits (16) and the final pair of digits (81) are squares.

1717:

a Pentagonal Number

1729:

a Carmichael Number; also a Dodecagonal Number

1820:

a Pentagonal Number, and 16C4

1926:

a Pentagonal Number

2035:

a Pentagonal Number

2147:

a Pentagonal Number

2262:

a Pentagonal Number

2035:

a Pentagonal Number, and 17C4

2449:

the 2449th Triangular Number is also a Dodecagonal Number

2501:

a Pentagonal Number, squares for first pair of digits, third, and fourth digit

2584:

the square root of a Square Dodecagonal Number

2625:

a Pentagonal Number

2752:

a Pentagonal Number

2882:

a Palindromic Pentagonal Number

3015:

a Pentagonal Number, with first two digits double its last two

3151:

a Pentagonal Number

3290:

a Pentagonal Number

3432:

a Pentagonal Number, and the 7th Central Binomial Coefficient

3577:

a Pentagonal Number, and a Kaprekar Constant in base 2

3725:

a Pentagonal Number

3876:

a Pentagonal Number

4030:

a Pentagonal Number, an an Abundant Number that is not the sum of some subset of its divisors

4187:

a Pentagonal Number, and the smallest Rabin-Miller Pseudoprime with an odd reciprocal period

4421:

1! - 2! + 3! - 4! + 5! - 6! + 7!

4676:

a Pentagonal Number, and the sum of the first seven 4th powers

4510:

a Pentagonal Number, and 444 + 55 + 11 + 0

4845:

a Pentagonal Number, and 20C4

5000:

Five Thousand One Hundred Four is the largest integer whose English name repeats no letter. See Numbers in Recreational Linguistics

5017:

a Pentagonal Number, and both 5 and 7 are Fermat Primes

5167:

1! + 3! + 5! + 7!

5192:

a Pentagonal Number

5370:

a Pentagonal Number

5551:

a Pentagonal Number

5735:

a Pentagonal Number

5922:

a Pentagonal Number

6112:

a Pentagonal Number

6305:

a Pentagonal Number

6501:

a Pentagonal Number, and has a square whose reverse is also a square

6623:

the 6623rd Dodecagonal Number is also a Triangular Number

6700:

a Pentagonal Number

6902:

a Pentagonal Number

6930:

the square root of a Square Triangular number (see Sloan A001110)

7107:

a Pentagonal Number

7315:

a Pentagonal Number, and 22C4

7526:

a Pentagonal Number

7740:

a Pentagonal Number

7921:

the 7921st Dodecagonal Number is also a Square

7957:

a Pentagonal Number

8177:

a Pentagonal Number

8400:

a Pentagonal Number, and divisible by its reverse

8626:

a Pentagonal Number

8855:

a Pentagonal Number, and 23C4

8911:

a Carmichael Number; also a Triangular Number, and Dodecagonal Number

9087:

a Pentagonal Number

9322:

a Pentagonal Number

9560:

a Pentagonal Number

9801:

a Pentagonal Number, and 9 times its reverse

10001:

10001 = 73 x 137 = 65^2 + 76^2.

Eric Weisstein's World of Mathematics

The home page pointing to (and searchable on) thousands of pages in the single most important Mathematics Site on the World Wide Web. Pi at Mathworld Mathematics in Film Mathematics in Literature

Hotlinks to Some Other Pages of Jonathan Vos Post

There are roughly 800 web pages in the doman created and run by Jonathan and his Physics Professor wife. The home page is:

Magic Dragon Multimedia

Some pages within that domain, which gets over 1,000,000 hits per month, include:

Software/Management Resume of Jonathan Vos Post

biographical and bibliographical info on Jonathan Vos Post PERIODIC TABLE OF MYSTERY AUTHORS

AUTHORS of Ultimate Science Fiction Web Guide

9,000+ more authors indexed

AUTHORS of Ultimate Westerns Web Guide

1,000+ more authors indexed

AUTHORS of Ultimate Romance Web Guide

8,000+ more authors indexed

A Discussion of Some Deeper Mathematical Issues Related to the Four Nines Puzzle

There is, of course, no upper limit to the numbers which we can build with nines and the operators mentioned at the top of this web page. Consider the infinite series: 9 9! (9!)! ((9!)!)! (((9!)!)!)! ((((9!)!)!)!)! ... and that just uses one 9. Playing around with the puzzle, it soon becomes obvious that there are integers that can be represented by four nines in an infinite number of different ways. It is NOT obvious whether ALL numbers can be represented in at least one way. We return to this later. By more advanced mathematics, it might be shown that every integer can be represented by a sufficiently long sequence of the operators we use here.

Dudeney Invented the Four Nines; Knuth Criticizes Dudeney

In "The Weekly Dispatch" of 4 February 1900, the puzzle column by Dudeney introduced this problem. But Professor Donald Knuth comments on Dudeney's Solution Number 310, which gives a table. Knuth criticizes: "he disallows (sqrt 9)! for completely illogical reasons; also, he fails to express 38, 41, 43, ... with fewer than five 9s." Knuth on Dudeney

Definition of Factorial

Let us note that there are some unsolved mathematical questions about the Factorial Function N! The well-known definition is: 1! = 1 2! = 1 x 2 = 2 3! = 1 x 2 x 3 = 6 4! = 1 x 2 x 3 x 4 = 24 5! = 1 x 2 x 3 x 4 x 5 = 120 6! = 1 x 2 x 3 x 4 x 5 x 6 = 720 7! = 1 x 2 x 3 x 4 x 5 x 6 x 7 = 5040 8! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 = 40,320 9! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 = 362,880 10! =1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 = 3,628,800 ... and so on

Any Integer is the First Digits of Some Factorial

Believe it or not, your telephone number, your Social Security Number, or any other whole number important to you can be found as the beginning digits of some enormous factorial. John E. Maxfield proved this theorem: "If A is any positive integer having M digits, there exists a positive integer N such that the first M digits of N! constitute the integer A." ["A Note on N!", John E. Maxfield. Kansas State University, Mathematics Magazine, March-April 1970, pp.64-67; supported in part by a National Science Foundation grant].

Factorials of Negative, Fractional, and Complex Numbers

For non-integers, and for complex numbers, the universally accepted generalization of the Factorial Function is the Gamma Function. Discussion of that shall be reserved until I provide my analysis of the Four Nines Puzzle as applied to Complex Numbers, such as first appear with sqrt (-9) = 3i.

Square-less-one Factorials

In our solutions to the Four Nines Puzzle, we often add to or subtract something from a factorial, and then take a square root. Typically, this is not a whole number, so we round down, or round up (floor or ceiling). This leads to the question: When is N! + 1 a perfect square? Or, equivalently, When is sqrt (N! + 1) a whole number? The only solutions known are: 25 = 4! + 1 = 5 x 5 121 = 5! + 1 = 11 x 11 5041 = 7! + 1 = 71 x 71 Nobody knows if there are any other perfect squares one more than factorials. Clearly related is a paper by Berend and Osgood: [Journal of Number Theory, vol. 42, 1992] which proves that for any polynomial P of degree > 1 the set of positive integers N for which P(X) = N! has an integral solution X, is of zero density. This paper explicitly says that it is not known if the equation X^2 - 1 = N! has only finitely many solutions. We also don't know if there are an infinite number of primes of the form n! + 1. We also don't know if there are an infinite number of primes of the form n! - 1. The largest such prime that we know is 3610! - 1. It has 11,277 digits [Caldwell, title to be added here, 1993] By the way, there is a fairly elementary proof that, except for 0!=1 and 1!=1, NO factorials are perfect squares. That fairly elementary proof, though, uses a heavy mathematical result known as Bertrand's Postulate, also known as Chebychev's Theorem (after the man who first proved it). This Theorem says that there always exists at least one prime between N and 2N, if N>2. Erdos gave a genuinely elementary (although neither short nor obvious) proof. This will all be inserted here or linked to in a later version of this web page.

Ergodic Hypothesis

The Four Nines Puzzle itself is, to be sure, very elementary stuff. So I stand meekly in the shadow of the great mathematician's who were my Teachers' teachers' teachers... Gottfried Leibnitz, Jacob Bernoulli, Johann Bernoulli, Leonhard Euler, Joseph Louis Lagrange, Simeon Poisson, Pafnuty Lvovich Chebyshev, Andrei A. Markov, G. H. Hardy, Alonzo Church, David Hilbert, Norbert Wiener, Alan Turing... With them in mind, I remark that Donald Knuth conjectures that ALL integers can be made with a sufficiently lengthy combination of square roots and factorials and floors and ceilings ... built around a single 4. I make the related conjecture, based on the number 9. As we see near the top of this web page, we can make a 4 from a single 9, with a lot of square roots, factorials, and floor functions. Hence Knuth's conjecture for 4 immediately applies to 9. In summary of a subtle proof of Knuth's Conjecture, still in progress, factorials make a number bigger, and square roots make it smaller. Iterating sufficiently, we are "folding" the algebraic number line back onto itself recursively, and this is an ergodic property, which carries a number arbitrarily close to any given integer, at which point a final floor or ceiling gets us exactly to that given integer.

Complexity Ordering of Solutions

The problem becomes (as I shall show in a forthcoming paper co-authored by Andrew Carmichael Post and Dr. George Hockney) how to achieve as many integers as possible with 4 nines -- for a given degree of "complexity" as defined by the number of symbols of a standardized way of expressing the combination of 9, 99, 999, 9999, +, -, x, /, sqrt, factorial, floor, and ceiling (say in Backus-Naur Form). For example: 9999 = 9999 (complexity = 4) 1 = 99/99 (complexity = 5) 1008 = 999 + 9 (complexity = 5) 9801 = 99 x 99 (complexity = 5) 2 = 99/9 - 9 (complexity = 6) 20 = 99/9 + 9 (complexity = 6) 19 = 9 + 9 + 9/9 (complexity 7) 36 = 9 + 9 + 9 + 9 (complexity 7) 13 = 9 + sqrt 9 + 9/9 (complexity 8) 40 = |_ sqrt 999 _| + 9 (complexity 8) and so on. The complexity function creates an order on the solutions to the four nines problem. Of interest are such functions as the smallest number whose complexity exceeds a given value, and upper and lower bounds on the ratio of a number to its complexity. Almost all numbers have very high complexity. But details will be revealed in that forthcoming paper. The problem of whether two strings of characters evaluate to the same integer is a very hard problem, in terms of the amount of computation necessary to determine it in general, called the "word problem" in complexity theory.

The "Four Nines Problem" is closely related to the "Four Fours Problem"

The "Four Fours Problem" first appeared in: "Mathematical Recreations and Essays", by W. W. Rouse Ball [1892]. In this book the "Four Fours Problem" is called a "traditional recreation." There are several fine sites on the World Wide Web for "Four Fours Problem." But I recommend to the reader: "Mathematical Games", by Martin Gardner, [Scientific America, Jan 1964]. The [Feb 1964] issue has answers to the puzzles posed in January. Martin Gardner was extending the "two fours" problem as first posed by J. A. Tierney in 1944, and extended by others in 1964. ["E631", J. A. Tierney, Amer. Math Monthly, 52(1945)219]. ["64 Ways to Write 64 Using four 4's", M. Bicknell and V. E. Hoggatt, Recreational Mathematics Magazine, 14(1964)13-25]. More recently, we have Knuth's Conjecture: "Representing Numbers Using Only One 4", Donald Knuth, [Mathematics Magazine, Vol. 37, Nov/Dec 1964, pp.308-310]. Knuth shows how (using a computer program he wrote) all integers from 1 through 207 may be represented with only one 4, varying numbers of square roots, varying numbers of factorials, and the floor function. For example: Knuth shows how to make the number 64 using only one 4: |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt |_ sqrt |_ sqrt sqrt sqrt sqrt sqrt (4!)! _| ! _| ! _| ! _| ! _| ! _| ! _| ! _| As to notation in the above example, he means sqrt n! stands for sqrt (n!), not (sqrt n)! Knuth further points out that |_ sqrt |_ X _| _| = |_ sqrt X _| so that the floor function's brackets are only needed around the entire result and before factorials are taken. He CONJECTURES that all integers may be represented that way: "It seems plausible that all positive integers possess such a representation, but this fact (if true) seems to be tied up with very deep propertis of the integers." Your Humble Webmaster believes that Knuth is right, for 9 as well as 4, and will prove that in a forthcoming paper. Knuth comments: "The referee has suggested a stronger conjecture, that a representation may be found in which all factorial operations precede all square root operations; and, moreover, if the greatest integer function [our floor function] is not used, an arbitrary positive real number can probably be approximated as closely as desired in this manner." If we abbreviate the long strings of symbols, we can express this more elegantly. Let NK be the number (...((4!)!)!...)! with K factorial operations. Then the anonymous referee's stronger conjecdture is equivalent to: log NK/(2^|_log(log NK)_|, for K=1,2,3..., are dense in the interval (1,2). Notationally here log means log to the base 2, and ^ means exponentiation. One key to the proof I shall publish is that, for any logarithm base: "The fractional part of log N is dense on the unit interval." From that, it has been proved that: "The fractional part of log N! is dense on [0,1]." ["A Note on N!", John E. Maxfield. Kansas State University, Mathematics Magazine, March-April 1970, pp.64-67; supported in part by a National Science Foundation grant]. Maxfield also proves, in the notation above, if we also define logK is the Kth iterant of Log N, i.e. log2(N) = log log N, that: The fractional part of logK (NK) is dense on the unit interval.

Square Root of a Sum of Square Roots

In our solutions of the Four Nines Puzzle, we sometimes take a square root of the sum of two slightly different square roots. As noted in Mathpages Mathpages #305: How can we find integer solutions of M = sqrt ( sqrt (N) + sqrt ((KxN)+1)) "This can be viewed as a Pell Equation with an extra solution on the solution." We have: M^2 = sqrt (N) + sqrt ((KxN)+1) in integers, so we know that there exist integers X and Y such that: N = Y^2 KN + 1 = X^2 X + Y = M^2 Eliminating N from the first two equations above gives the Pell equation: X^2 + KY^2 = 1 For any given K we are looking for solutions X,Y such that X+Y is a square. "Of course, for any positive integer K there are infinitely many solutions to the Pell Equation, but solutions with X+Y=square are rare. For example, with K=8, the values of X+Y satisfying the Pell Equation" are: ((16 + 11 sqrt 2)/8) (3 + sqrt 8)^Q + ((16 - 11 sqrt 2)/8) (3 - sqrt 8)^Q This gives the sequence 4, 23, 134, 781, 4552 ..., which satisfies the second-order recurrence: S(j) = 6S(j-1) - S(j-2) "So the question is whether this sequence contains any squares after the initial vale 4. Recall the proof that the only square Fibonacci numbers are 0, 1, and 144." [J. Cohn, "On the Square Fibonacci Numbers", J. London Math Soc., 39 (1964) 537-540] "In general the problem reduces to finding square terms of a general second-order recurring sequence, like the Fibonacci sequence. The best approach might be to apply Cohn's method of proof to the general second-order recurrence."

Square Roots of Factorials in Quantum Mechanics

In our solutions to the "Four Nines Problem" we often take square roots of factorials, and sometimes square roots of square roots of factorial of factorials, and so forth. As it turns out, there are some solved and some unsolved problems in the Physics of Quantum Wave Functions which involve square roots of factorials. As B. Nagel has written, "Mathematical problems in Quantum Optics", {ref to be done}: "As a continuation of earlier studies of squeezed states and other special harmonic oscillator states of interest in quantum optics I have studied the phase representations of these states. Although the phase observable, which is roughly speaking conjugate to the number operator, does not exist as a hermitian operator -- this is a longstanding and still popular problem, initiated by Dirac in 1927 -- it exists as a so-called general observable and permits a probability interpretation via a phase distribution on the unit circle. The corresponding wave function is obtained simply by substituting the harmonic wave exp(in[[phi]]) for the number state |n>. The series expansions thus obtained for the coherent and squeezed states contain a square root of a factorial n!, which makes it impossible to get closed analytical expressions. Approximate analytical expressions have been derived, valid for large values of the mean value of the number operator...." For another example, see the computer program given in: Grozin Hydrogen Wave Functions and E1 Transitions Procedure R(n,1); % radial wave function -2/n^2*sqrt(factorial(n+1)/factorial(n-1-1))*exp(-r/n) ... etc. Wouldn't it be interesting if an in-depth analysis of the century-old recreation math puzzle about Four Nines turned out to be useful in solving a problem in 21st Century Quantum Optics with Squeezed States? {more discussion to be added in February 2004} Special Thanks to Dr. George Hockney, NASA/JPL, for informal discussion and review in January-February 2004. Thanks to Forrest Bishop for informal discussion and review in January-February 2004.
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