[FOM] 437: Twin Prime Polynomial
Harvey Friedman
friedman at math.ohio-state.edu
Wed Sep 15 13:09:46 EDT 2010
THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION.
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INTEGER POLYNOMIAL DECISION PROBLEMS
September 15, 2010
The following result illustrates the difficulties inherent in
understanding the set of nonnegative values of rather simple
polynomials from N^3 into Z.
THEOREM. The Twin Prime Conjecture is equivalent to the assertion that
a particular polynomial P:N^3 into Z of degree 4 omits infinitely many
nonnegative values. Furthermore, the coefficients are 1,2,4,6,-1,-2.
To see this, note that
k is not a twin prime iff
1. (forall x,y >= 0)(k = (x+2)(y+2) or k = (x+2)(y+2)+2)
2. (forall x,y,z >= 0)(k = (x+2)(y+2)+z and z(z-2) = 0)
3. (forall x,y,z >= 0)(k = (x+2)(y+2)+2z and z(1-z)+1 = 1)
4. (forall x,y,z >= 0)(k = ((x+2)(y+2)+2z)(z(1-z)+1))
Set P(x) = ((x+2)(y+2)+2z)(z(1-z)+1)) = (xy+2x+2y+2z+4)(-z^2+z+1) =
-xyz^2+xyz+xy-2xz^2+2xz+2x-2yz^2+2yz+2y-2z^3+2z^2+2z-4z^2+4z+4 =
-xyz^2+xyz+xy-2xz^2+2xz+2x-2yz^2+2yz+2y-2z^3-2z^2+6z+4. QED
PROBLEM: Can we find a "simpler" polynomial for the above Theorem?
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I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 437th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-349 can be found athttp://www.cs.nyu.edu/pipermail/fom/2009-August/014004.html
in the FOM archives.
350: one dimensional set series 7/23/09 12:11AM
351: Mapping Theorems/Mahlo/Subtle 8/6/09 10:59PM
352: Mapping Theorems/simpler 8/7/09 10:06PM
353: Function Generation 1 8/9/09 12:09PM
354: Mahlo Cardinals in HIGH SCHOOL 1 8/9/09 6:37PM
355: Mahlo Cardinals in HIGH SCHOOL 2 8/10/09 6:18PM
356: Simplified HIGH SCHOOL and Mapping Theorem 8/14/09 9:31AM
357: HIGH SCHOOL Games/Update 8/20/09 10:42AM
358: clearer statements of HIGH SCHOOL Games 8/23/09 2:42AM
359: finite two person HIGH SCHOOL games 8/24/09 1:28PM
360: Finite Linear/Limited Memory Games 8/31/09 5:43PM
361: Finite Promise Games 9/2/09 7:04AM
362: Simplest Order Invariant Game 9/7/09 11:08AM
363: Greedy Function Games/Largest Cardinals 1
364: Anticipation Function Games/Largest Cardinals/Simplified 9/7/09
11:18AM
365: Free Reductions and Large Cardinals 1 9/24/09 1:06PM
366: Free Reductions and Large Cardinals/polished 9/28/09 2:19PM
367: Upper Shift Fixed Points and Large Cardinals 10/4/09 2:44PM
368: Upper Shift Fixed Point and Large Cardinals/correction 10/6/09
8:15PM
369. Fixed Points and Large Cardinals/restatement 10/29/09 2:23PM
370: Upper Shift Fixed Points, Sequences, Games, and Large Cardinals
11/19/09 12:14PM
371: Vector Reduction and Large Cardinals 11/21/09 1:34AM
372: Maximal Lower Chains, Vector Reduction, and Large Cardinals
11/26/09 5:05AM
373: Upper Shifts, Greedy Chains, Vector Reduction, and Large
Cardinals 12/7/09 9:17AM
374: Upper Shift Greedy Chain Games 12/12/09 5:56AM
375: Upper Shift Clique Games and Large Cardinals 1graham
376: The Upper Shift Greedy Clique Theorem, and Large Cardinals
12/24/09 2:23PM
377: The Polynomial Shift Theorem 12/25/09 2:39PM
378: Upper Shift Clique Sequences and Large Cardinals 12/25/09 2:41PM
379: Greedy Sets and Huge Cardinals 1
380: More Polynomial Shift Theorems 12/28/09 7:06AM
381: Trigonometric Shift Theorem 12/29/09 11:25AM
382: Upper Shift Greedy Cliques and Large Cardinals 12/30/09 2:51AM
383: Upper Shift Greedy Clique Sequences and Large Cardinals 1
12/30/09 3:25PM
384: THe Polynomial Shift Translation Theorem/CORRECTION 12/31/09
7:51PM
385: Shifts and Extreme Greedy Clique Sequences 1/1/10 7:35PM
386: Terrifically and Extremely Long Finite Sequences 1/1/10 7:35PM
387: Better Polynomial Shift Translation/typos 1/6/10 10:41PM
388: Goedel's Second Again/definitive? 1/7/10 11:06AM
389: Finite Games, Vector Reduction, and Large Cardinals 1 2/9/10
3:32PM
390: Finite Games, Vector Reduction, and Large Cardinals 2 2/14/09
10:27PM
391: Finite Games, Vector Reduction, and Large Cardinals 3 2/21/10
5:54AM
392: Finite Games, Vector Reduction, and Large Cardinals 4 2/22/10
9:15AM
393: Finite Games, Vector Reduction, and Large Cardinals 5 2/22/10
3:50AM
394: Free Reduction Theory 1 3/2/10 7:30PM
395: Free Reduction Theory 2 3/7/10 5:41PM
396: Free Reduction Theory 3 3/7/10 11:30PM
397: Free Reduction Theory 4 3/8/10 9:05AM
398: New Free Reduction Theory 1 3/10/10 5:26AM
399: New Free Reduction Theory 2 3/12/10 9:36AM
400: New Free Reduction Theory 3 3/14/10 11:55AM
401: New Free Reduction Theory 4 3/15/10 4:12PM
402: New Free Reduction Theory 5 3/19/10 12:59PM
403: Set Equation Tower Theory 1 3/22/10 2:45PM
404: Set Equation Tower Theory 2 3/24/10 11:18PM
405: Some Countable Model Theory 1 3/24/10 11:20PM
406: Set Equation Tower Theory 3 3/25/10 6:24PM
407: Kernel Tower Theory 1 3/31/10 12:02PM
408: Kernel tower Theory 2 4/1/10 6:46PM
409: Kernel Tower Theory 3 4/5/10 4:04PM
410: Kernel Function Theory 1 4/8/10 7:39PM
411: Free Generation Theory 1 4/13/10 2:55PM
412: Local Basis Construction Theory 1 4/17/10 11:23PM
413: Local Basis Construction Theory 2 4/20/10 1:51PM
414: Integer Decomposition Theory 4/23/10 12:45PM
415: Integer Decomposition Theory 2 4/24/10 3:49PM
416: Integer Decomposition Theory 3 4/26/10 7:04PM
417: Integer Decomposition Theory 4 4/28/10 6:25PM
418: Integer Decomposition Theory 5 4/29/10 4:08PM
419: Integer Decomposition Theory 6 5/4/10 10:39PM
420: Reduction Function Theory 1 5/17/10 2:53AM
421: Reduction Function Theory 2 5/19/10 12:00PM
422: Well Behaved Reduction Functions 1 5/23/10 4:12PM
423: Well Behaved Reduction Functions 2 5/27/10 3:01PM
424: Well Behaved Reduction Functions 3 5/29/10 8:06PM
425: Well Behaved Reduction Functions 4 5/31/10 5:05PM
426: Well Behaved Reduction Functions 5 6/2/10 12:43PM
427: Finite Games and Incompleteness 1 6/10/10 4:08PM
428: Typo Correction in #427 6/11/10 12:11AM
429: Finite Games and Incompleteness 2 6/16/10 7:26PM
430: Finite Games and Incompleteness 3 6/18/10 6:14PM
431: Finite Incompleteness/Combinatorially Simplest 6/20/10 11:22PM
432: Finite Games and Incompleteness 4 6/26/10 8:39PM
433: Finite Games and Incompleteness 5 6/27/10 3:33PM
434: Digraph Kernel Structure Theory 1 7/4/10 3:17PM
435: Kernel Structure Theory 1 7/5/10 5:55PM
436: Kernel Structure Theory 2 7/9/10 5:21PM
Harvey Friedman
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