December 2011 - mathfun@mailman.xmission.com

[math-fun] OEIS at AMS meeting in Boston in January
by N. J. A. Sloane 16 Dec '11

16 Dec '11

Dear Math Fun, If any of you are attending the annual AMS/MAA meeting in Boston (Jan 3-7), I'm talking at 1pm - 1:30pm on Fri Jan 6 in Hynes Convention Center, Room 204 (on Dismal Arithmetic and the OEIS). I'm staying at the Sheraton Boston Hotel, 39 Dalton St, 617 236 2000. I also propose that fans of the OEIS meet for dinner at 7pm on Fri Jan 6 in the lobby of the Sheraton - let me know if you are interested. Best regards Neil

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[math-fun] Optimum algorithms for generating Heron triangles; better count bounds & generators for d-simplices
by Warren Smith 16 Dec '11

16 Dec '11

Optimum algorithms for generating Heron triangles; better count bounds & generators for d-simplices ====================Warren=D=Smith======Dec=2011====================================== I wondered if there were an algorithm to generate Heron triangles with diameter<N in N^(1+o(1)) time, which would be optimum. ["diameter" = longest edge. "Heron" = all sides, and face areas, and ..., volume are integer.] Sascha Kurz: On the generation of Heronian triangles, Serdica J. Comput. 2,2 (2008) 181-196 gave several algorithms for the triangle-generation task that run in N^(2+o(1)) time. We will now explain how to 1. Generate Heron triangles with diameter=N in N^o(1) time, and essentially no space. 2. Hence can generate Heron triangles with diameter<=N in N^(1+o(1)) time, and essentially no space. 3. Hence can generate Heron tetrahedra with diameter=N in N^o(1) time, and essentially no space. 4. Hence can generate Heron tetrahedra with diameter<=N in N^(1+o(1)) time, and essentially no space. Hence there are N^(1+o(1)) such tetrahedra, a new superior counting upper bound. 5. This keeps going to 4D, 5D, etc; works in any fixed dimension. PROOF: Let F(n) be the number of Heron triangles with diameter=n. IF we can prove an upper bound F(n)<n^o(1) it then would immediately follow that the number of Heron tetrahedra and Heron triangles with diameter<N is N^(1+o(1)). INCENTIVIZING REMARK: It is clear from previous bounds by me that F(N) has average order N^o(1), but we need to bound its maximum order. INCENTIVIZING REMARK: We know (a related problem) that the number of Pythagorean right triangles with hypotenuse C is C^o(1), in fact it is bounded by something like 4^G where G is the number of Gaussian primes in the factorization of C [there is actually a known formula for this count in terms of prime factorization of C, and there is a known fast algorithm dating to Hermite, also given by D.Shanks in his number theory book, also I think in Bach+Shallit's book, for finding all the representations of a given number as sum of two squares]. PROOF CONTINUES: In the Heron/Brahmagupta formulas for the side lengths a,b,c of a Heron triangle a = (n*n + k*k)*m; b = (m*m + k*k)*n; c = (m + n)*(m*n - k*k); we can argue the #divisors and factorizations a=X*Y of a is a^o(1) at most [Wigert bound] and the number of representations of a number X as sum of two squares like X=n*n+k*k is X^o(1) at most [Gaussian primes bound] hence if a is known the number of possible (k,m,n) it could have come from, is a^o(1) at most and we can find them all fast using fast integer and Gaussian integer factorization algorithms such as Dixon's rigorous "quadratic sieve"-like randomized factorizing algorithm. Similarly, if b is known, the number of possible (k,m,n) it could have come from, is b^o(1) at most. If c is known, it is a little trickier. We can factorize it into 2 factors in c^o(1) ways at most, c=X*Y, where X=m+n Y=m*n-k*k. If X and Y are known, then we can solve for n and m in terms of {X,Y,k} to find n,m = X/2 +- sqrt(X*X-4*Y-4*k*k)/2 hence the number of integer solutions n,m is bounded in terms of the number of ways that X*X-4*Y-4*k*k can be a square s*s. That is... in terms of the number of representations of X*X-4*Y = 4*k*k+s*s = (2*k)^2 + s^2 as a sum of two squares! But we already showed this is (X*X-4*Y)^o(1) <= c^o(1). Hence THEOREM: The number F(n) of Heron triangles with diameter=n, is F(n)<n^o(1), and they can be generated from n in n^o(1) time. RELATED WEAKER EARLIER RESULT: E.J. Ionascu, F. Luca, P.Stanica: Heron triangles with two fixed sides, J. Number Theory 126,1 (2007) 52-67. http://faculty.nps.edu/pstanica/research/heron.pdf studies the function H(a,b), which associates to every pair of positive integers a and b the number of positive integers c such that the triangle of sides a, b and c is Heron, i.e, has integral area. They prove H(a,b)<=4*NumberOfDivisors(a*b) and their method implicitly yields a way to generate all the c's from a,b in (a*b)^o(1) time. COROLLARY: The number of Heron tetrahedra with diameter=n, is <F(n)*F(n)*n^o(1)=n^o(1) and they can be generated from n in n^o(1) time. These constitute generation algorithms for Heron triangles with diameter<N that run in optimum time N^(1+o(1)) [well, optimum up to the vagueness in the "o(1)"] and optimum space [essentially no space]. It might however be that for Heron tetrahedra this is non-optimum. The question depends on how many Heron tetrahedra there are. This all gives us a new upper bound N^(1+o(1)) on the number of Heron tetrahedra of diameter<N. It also gives us an upper bound N^(1+o(1)) on the number of Heron triangles of diameter<N in a far simpler manner than my previous proof of that. But it might be that an as-yet-unknown better upper bound such as N^0.9 holds in which case our generator might not be optimum for tetrahedra. COROLLARY: The number of Heron 4-simplices with diameter=n, is <F(n)*F(n)*F(n)*n^(3*o(1))=n^o(1) and they can be generated from n in n^o(1) time. The three F(n)'s come from the 3 triangles sharing the longest edge, the n^(3*o(1)) come from the 3 simplex edges not in those 3 triangles. And so on... COROLLARY: In any fixed dimension d>0, the number of Heron d-simplices with diameter=n, is <F(n)^(d-1)*n^(((d+1)*d/2-2*d+1)*o(1))=n^o(1) and they can be generated from n in n^o(1) time. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] Puzzler
by Michael Reid 16 Dec '11

16 Dec '11

> I give you an arbitrary positive integer x > 0. > > You write down x, 2x, 3x, 4x, ..., kx in decimal. > > How large much k be to guarantee that every digit appears somewhere in > your list, regardless of x? What is the smallest k for which there are k positive integers m_1 , m_2 , ... , m_k such that, for every positive integer x , all ten decimal digits occur on the list m_1 x , m_2 x , ... , m_k x ? (The solution of the original problem shows that such k exist.) Michael Reid

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[math-fun] Puzzler
by David Wilson 15 Dec '11

15 Dec '11

I give you an arbitrary positive integer x > 0. You write down x, 2x, 3x, 4x, ..., kx in decimal. How large much k be to guarantee that every digit appears somewhere in your list, regardless of x?

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[math-fun] A Xmas present
by Eric Angelini 15 Dec '11

15 Dec '11

Be the star of the evening (at least with the youngsters). Print this enumeration on a few sheets of paper and give pencils around: Six, one, eight, five, two, seven, four, ten, one, two, two, three, seven, zero, close. Ask the kids to draw a thin vertical stroke after each letter "e". Count the letters between the strokes. Enjoy! Best, É.

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[math-fun] The 394 primitive Heron tetrahedra with diameter<=16384
by Warren Smith 15 Dec '11

15 Dec '11

Generated by Warren D. Smith & Fred Lunnon, December 2011. "Heron" tetrahedra = all sides, face-areas, and volume = integers. "Primitive" = gcd(sidelengths)=1. [We proved the theorem that volume is always divisible by 168 but this is not true for any divisor greater than 168.] [A much larger list of Heron tetrahedra, up to diameter 70000, was constructed by Sascha Kurz but a bug in his program caused a small fraction of omissions -- 4%?.] Census: A=[diameter] B=[#primtets with diam<=A] C=[exponent in power law estimated from this & preceding count] A B C -------------------------------- 64 0 128 1 256 4 2.000 512 9 1.170 1024 27 1.585 2048 62 1.199 4096 129 1.057 8192 218 0.757 16384 394 0.854 [s01, s02, s12, s03, s13, s23] ------------------------------------------------- [117, 84, 51, 80, 53, 52], [160, 153, 25, 120, 56, 39], [203, 195, 148, 148, 195, 203], [225, 200, 65, 87, 156, 119], [318, 221, 203, 221, 175, 42], [319, 318, 175, 221, 210, 175], [319, 318, 175, 221, 252, 203], [429, 300, 261, 176, 275, 140], [468, 340, 232, 297, 225, 65], [595, 429, 208, 325, 276, 116], [595, 507, 116, 325, 276, 208], [595, 555, 100, 429, 208, 204], [612, 480, 156, 455, 319, 185], [671, 663, 580, 225, 544, 444], [680, 672, 104, 615, 185, 153], [680, 680, 208, 615, 185, 185], [697, 680, 153, 672, 185, 104], [697, 697, 306, 672, 185, 185], [741, 477, 420, 475, 304, 340], [780, 765, 219, 715, 221, 148], [884, 880, 84, 715, 205, 187], [884, 884, 168, 715, 205, 205], [888, 875, 533, 533, 875, 888], [925, 765, 520, 756, 533, 117], [935, 928, 345, 900, 145, 260], [990, 795, 663, 598, 436, 427], [990, 901, 793, 793, 901, 308], [1040, 925, 925, 533, 533, 756], [1073, 975, 448, 495, 952, 840], [1073, 1071, 520, 975, 448, 264], [1073, 1073, 896, 495, 952, 952], [1073, 1073, 896, 1071, 520, 520], [1073, 1073, 990, 448, 975, 975], [1073, 1073, 990, 896, 1073, 1073], [1105, 1073, 264, 952, 561, 495], [1105, 1073, 264, 975, 520, 448], [1105, 1105, 528, 952, 561, 561], [1105, 1105, 528, 975, 520, 520], [1113, 975, 348, 625, 572, 400], [1160, 1125, 595, 820, 660, 793], [1190, 1056, 218, 435, 925, 819], [1248, 1155, 765, 1040, 1040, 445], [1276, 924, 880, 765, 533, 555], [1344, 697, 697, 680, 680, 153], [1344, 697, 697, 697, 697, 306], [1358, 1221, 445, 654, 1040, 765], [1360, 1092, 548, 975, 865, 663], [1450, 1184, 1066, 825, 1025, 697], [1480, 1275, 1189, 1248, 1160, 261], [1480, 1275, 1189, 1275, 1189, 522], [1500, 1184, 572, 825, 975, 697], [1512, 925, 925, 765, 765, 520], [1512, 925, 925, 925, 925, 1040], [1624, 1575, 455, 1560, 1184, 825], [1769, 1480, 969, 1066, 1197, 570], [1804, 1479, 1183, 1183, 1479, 1804], [1885, 1258, 1001, 867, 1192, 425], [1904, 1105, 1105, 1073, 1073, 264], [1904, 1105, 1105, 1105, 1105, 528], [1904, 1813, 1365, 1300, 1332, 957], [1950, 1105, 1105, 1073, 1073, 264], [1950, 1105, 1105, 1105, 1105, 528], [2067, 2040, 333, 1732, 725, 644], [2067, 2067, 666, 1732, 725, 725], [2109, 1435, 884, 1189, 1300, 624], [2165, 2067, 644, 2040, 725, 333], [2165, 2165, 1288, 2040, 725, 725], [2167, 1590, 1105, 1105, 1590, 875], [2167, 1980, 1595, 1590, 1105, 1050], [2175, 1820, 425, 1776, 1599, 1276], [2257, 2023, 900, 1820, 969, 273], [2320, 1560, 776, 1233, 1105, 399], [2380, 2079, 455, 1716, 1184, 825], [2384, 1845, 1547, 1020, 1628, 1635], [2431, 2296, 2175, 1479, 1700, 2375], [2431, 2296, 2175, 1540, 1881, 1092], [2431, 2296, 2175, 1700, 1479, 1044], [2431, 2296, 2175, 2175, 2296, 2431], [2431, 2375, 1044, 2175, 2296, 1700], [2431, 2375, 1044, 2296, 2175, 1479], [2652, 2640, 252, 533, 2275, 2261], [2652, 2652, 504, 533, 2275, 2275], [2664, 2600, 1504, 1025, 2431, 2175], [2664, 2600, 1504, 1599, 2175, 2431], [2664, 2625, 1599, 656, 2600, 2431], [2664, 2625, 1599, 1025, 2431, 2600], [2664, 2625, 1599, 2431, 1025, 656], [2664, 2625, 1599, 2600, 656, 1025], [2665, 2431, 1092, 1540, 2175, 1881], [2665, 2665, 2184, 1540, 2175, 2175], [2720, 2109, 1859, 884, 1908, 1435], [2720, 2405, 1435, 884, 1908, 1859], [2805, 2340, 861, 1360, 1547, 1036], [2873, 2748, 1825, 1825, 2748, 2873], [2912, 2379, 715, 1768, 1320, 715], [2928, 2500, 1508, 2275, 1507, 1275], [2960, 2665, 425, 1628, 1924, 1647], [3080, 2665, 2665, 2431, 2431, 1092], [3080, 2665, 2665, 2665, 2665, 2184], [3111, 2639, 2180, 2180, 2639, 3111], [3179, 2640, 1771, 1476, 2275, 1428], [3211, 2625, 2378, 2028, 2275, 1353], [3219, 3179, 700, 3060, 507, 493], [3276, 2860, 592, 1485, 1989, 1595], [3277, 2652, 1925, 2052, 2555, 1680], [3444, 3360, 756, 3179, 925, 533], [3444, 3444, 1512, 3179, 925, 925], [3485, 2652, 2261, 2640, 2275, 252], [3485, 2779, 1044, 2405, 1140, 816], [3485, 3179, 1428, 2640, 2275, 1771], [3485, 3444, 533, 3360, 925, 756], [3485, 3485, 1066, 3360, 925, 925], [3485, 3485, 2856, 2640, 2275, 2275], [3542, 3179, 3179, 2275, 2275, 1476], [3762, 2431, 2431, 2175, 2175, 2296], [3774, 3731, 475, 3650, 416, 549], [3825, 2639, 1424, 2535, 2460, 1924], [3825, 3770, 2555, 1914, 2373, 3248], [3825, 3780, 585, 1628, 2665, 2600], [3825, 3825, 1170, 1628, 2665, 2665], [3850, 3277, 3277, 2555, 2555, 2052], [3944, 3900, 740, 3825, 1105, 1095], [3965, 3875, 840, 1364, 3723, 3627], [3965, 3965, 1680, 1364, 3723, 3723], [3965, 3965, 2728, 840, 3875, 3875], [3965, 3965, 2728, 1680, 3965, 3965], [4000, 3649, 2847, 3460, 2340, 3549], [4080, 2165, 2165, 2067, 2067, 644], [4080, 2165, 2165, 2165, 2165, 1288], [4104, 3277, 3277, 2652, 2652, 1925], [4104, 3277, 3277, 3277, 3277, 3850], [4147, 4134, 2275, 2958, 2639, 3276], [4181, 4100, 819, 3740, 1869, 1680], [4181, 4181, 1638, 3740, 1869, 1869], [4200, 2793, 2583, 2015, 2465, 2260], [4200, 4147, 493, 3900, 540, 533], [4225, 4199, 468, 4180, 615, 399], [4225, 4225, 936, 4180, 615, 615], [4256, 3477, 1045, 3400, 1160, 533], [4264, 2405, 1989, 2072, 2400, 1221], [4264, 2405, 1989, 2400, 2072, 155], [4344, 3885, 3315, 3392, 3848, 1235], [4453, 3828, 2275, 2772, 3485, 2640], [4514, 3731, 1275, 3640, 1938, 819], [4522, 3485, 3485, 2275, 2275, 2640], [4550, 4453, 4453, 3485, 3485, 2772], [4753, 4100, 3757, 3757, 4100, 1737], [4784, 4641, 1375, 2484, 2668, 2535], [5100, 2929, 2821, 2910, 2670, 221], [5232, 4879, 4225, 1768, 3880, 4505], [5249, 4628, 1173, 4250, 1449, 690], [5280, 3485, 3485, 2652, 2652, 2261], [5280, 3485, 3485, 3179, 3179, 1428], [5280, 3485, 3485, 3485, 3485, 2856], [5280, 3485, 3485, 3485, 3485, 4522], [5365, 4760, 2475, 3952, 2805, 1320], [5365, 5365, 4950, 3952, 2805, 2805], [5423, 4305, 1976, 3145, 2652, 1300], [5432, 4539, 2405, 4080, 4264, 1989], [5491, 4715, 2340, 3059, 4560, 3588], [5500, 5355, 845, 5040, 5084, 5355], [5512, 5215, 1887, 1887, 5215, 5512], [5525, 4715, 2880, 4301, 3468, 1932], [5525, 4884, 1603, 3432, 2197, 1980], [5544, 4453, 4453, 3828, 3828, 2275], [5544, 4453, 4453, 4453, 4453, 4550], [5577, 4305, 2196, 3145, 2848, 1300], [5577, 5423, 220, 3145, 2848, 2652], [5760, 5525, 5525, 3468, 3468, 4301], [5920, 4875, 1397, 3485, 4125, 2860], [5936, 5100, 3916, 3731, 5075, 2405], [6005, 4947, 3404, 4653, 3796, 1680], [6118, 4575, 2873, 2873, 4575, 3368], [6118, 5491, 5491, 4715, 4715, 2340], [6120, 5551, 1199, 3825, 3825, 2824], [6292, 6273, 4375, 3828, 4400, 2775], [6327, 5423, 1300, 4305, 2652, 1976], [6369, 5576, 4225, 4225, 5576, 6111], [6448, 6095, 1073, 3731, 2925, 2788], [6516, 4920, 2436, 3325, 4901, 2813], [6625, 6409, 4116, 3116, 6409, 6625], [6699, 5100, 2691, 4172, 3325, 976], [6699, 6042, 1455, 3325, 5404, 4199], [6720, 3485, 3485, 3444, 3444, 533], [6720, 3485, 3485, 3485, 3485, 1066], [6808, 6005, 6005, 3796, 3796, 4653], [6851, 5565, 2000, 4875, 2860, 1740], [7011, 6699, 5100, 4579, 3800, 4628], [7059, 5330, 2639, 4845, 2784, 1073], [7085, 7076, 357, 6960, 1325, 1276], [7085, 7085, 714, 6960, 1325, 1325], [7105, 5512, 1887, 5180, 3465, 1788], [7215, 5577, 2652, 5423, 2848, 220], [7293, 6565, 4732, 3480, 5859, 4165], [7293, 6868, 4165, 3480, 5859, 4732], [7325, 7072, 1853, 6328, 1347, 1160], [7395, 7011, 2352, 2405, 5740, 5236], [7395, 7395, 4704, 2405, 5740, 5740], [7400, 6273, 2783, 6048, 4264, 1665], [7400, 6273, 5537, 6048, 4264, 1665], [7480, 4181, 4181, 4100, 4100, 819], [7480, 4181, 4181, 4181, 4181, 1638], [7480, 5624, 5200, 4427, 4053, 2603], [7585, 7400, 1665, 4264, 6273, 6048], [7585, 7585, 3330, 4264, 6273, 6273], [7667, 6396, 4879, 5220, 3185, 3696], [7667, 6396, 4879, 6396, 4879, 7392], [7735, 6327, 2848, 4305, 4900, 2652], [7735, 6327, 2848, 5423, 2924, 1300], [7735, 7215, 1300, 5423, 2924, 2848], [7888, 4732, 3620, 4575, 3487, 857], [7950, 7469, 5525, 6783, 3393, 2548], [8075, 7917, 4756, 7605, 2000, 5772], [8120, 7800, 5920, 4125, 6851, 7875], [8120, 7800, 5920, 6851, 4125, 2275], [8120, 7875, 2275, 5920, 7800, 6851], [8160, 7215, 4305, 2848, 5504, 5423], [8160, 7700, 1300, 5577, 6327, 5423], [8250, 7626, 7560, 6401, 6601, 7735], [8288, 6396, 5500, 5775, 4879, 6771], [8360, 4225, 4225, 4199, 4199, 468], [8360, 4225, 4225, 4225, 4225, 936], [8385, 8364, 4407, 6880, 2623, 2516], [8400, 7395, 5085, 3920, 4928, 3875], [8428, 8160, 1300, 4305, 7735, 7215], [8428, 8160, 1300, 7735, 4305, 3145], [8432, 6380, 3404, 3825, 4879, 2665], [8484, 6625, 6409, 6409, 6625, 8484], [8528, 7585, 7585, 7400, 7400, 1665], [8528, 7585, 7585, 7585, 7585, 3330], [8602, 5525, 5525, 4715, 4715, 2880], [8602, 5525, 5525, 5525, 5525, 5760], [8645, 7056, 3451, 5820, 3965, 4764], [8721, 8424, 3375, 1417, 8140, 7735], [8721, 8424, 3375, 7412, 7735, 8140], [8800, 7216, 6864, 3625, 6825, 4329], [8840, 8535, 7625, 5952, 6536, 3927], [8925, 7708, 2533, 7520, 1885, 1692], [8925, 7708, 2533, 7708, 2533, 3384], [9020, 7395, 5915, 6864, 5852, 861], [9020, 8925, 2405, 8736, 1564, 1827], [9020, 8925, 2405, 8925, 2405, 3654], [9061, 7395, 5236, 7011, 5740, 2352], [9061, 8925, 1564, 8736, 2405, 1827], [9061, 9020, 861, 6864, 5915, 5852], [9061, 9061, 1722, 6864, 5915, 5915], [9061, 9061, 3128, 8736, 2405, 2405], [9295, 8215, 5916, 4785, 7216, 4732], [9306, 6005, 6005, 4947, 4947, 3404], [9306, 6005, 6005, 6005, 6005, 6808], [9367, 8800, 4329, 7216, 6825, 6864], [9425, 7599, 2464, 7215, 4160, 3936], [9435, 8352, 4389, 6740, 5075, 2548], [9435, 9435, 8778, 6740, 5075, 5075], [9471, 7800, 2529, 5920, 4321, 4040], [9605, 9228, 6253, 7800, 5605, 2772], [9646, 9605, 2829, 6500, 4554, 5865], [9673, 8840, 3927, 5952, 7625, 6536], [9673, 9673, 7854, 5952, 7625, 7625], [9700, 8584, 4836, 8125, 2325, 3069], [9773, 9435, 2548, 8352, 5075, 4389], [9773, 9773, 5096, 8352, 5075, 5075], [9835, 9074, 4125, 5920, 6165, 3570], [9840, 8976, 4032, 3145, 7735, 6601], [9840, 9840, 8064, 3145, 7735, 7735], [9881, 8938, 3681, 4920, 8569, 7462], [9881, 9881, 9840, 3681, 8938, 8938], [9997, 9605, 2772, 7800, 6253, 5605], [9997, 9997, 5544, 7800, 6253, 6253], [10324, 9840, 3124, 6771, 5525, 4557], [10324, 10324, 6248, 6771, 5525, 5525], [10400, 7656, 3416, 7025, 3825, 689], [10441, 6409, 5250, 5250, 6409, 1691], [10455, 9052, 4255, 8784, 2409, 4732], [10472, 9061, 9061, 5740, 5740, 7011], [10624, 8160, 2848, 5577, 7735, 6327], [10624, 8160, 2848, 7735, 5577, 3145], [10647, 7735, 5278, 6327, 7650, 2848], [10665, 8432, 2665, 6380, 4879, 3404], [10693, 8772, 2465, 7120, 5307, 4060], [10736, 10275, 2509, 9724, 3300, 1099], [10920, 9090, 2670, 9061, 2405, 1015], [10959, 10400, 689, 7656, 3825, 3416], [11165, 9860, 2523, 9724, 4059, 1632], [11210, 9605, 9605, 6253, 6253, 9228], [11232, 10089, 4329, 6136, 8120, 5015], [11275, 10136, 8619, 8619, 10136, 11275], [11285, 10324, 4557, 9840, 5525, 3124], [11285, 11285, 9114, 9840, 5525, 5525], [11375, 9280, 2385, 6832, 6545, 5168], [11655, 7820, 6929, 7728, 6825, 1196], [11655, 7820, 6929, 7820, 6929, 2392], [11660, 10812, 5512, 9945, 5215, 1887], [11687, 10545, 5740, 9945, 5512, 1740], [11687, 10812, 5215, 9945, 5512, 1887], [11687, 11660, 1887, 9945, 5512, 5215], [11687, 11660, 1887, 10812, 5215, 5512], [11700, 8316, 4176, 7735, 4225, 3199], [11704, 9020, 9020, 5915, 5915, 7395], [11713, 11615, 1512, 10488, 5215, 4991], [11713, 11713, 3024, 10488, 5215, 5215], [11737, 10032, 7975, 9828, 4625, 3900], [11832, 11165, 1885, 5460, 6732, 6475], [11849, 9840, 6601, 8976, 7735, 4032], [11904, 9673, 9673, 8840, 8840, 3927], [11904, 9673, 9673, 9673, 9673, 7854], [11951, 11440, 2415, 9028, 4329, 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12155, 5460, 10373, 8364, 6336], [13325, 13243, 1476, 8645, 10140, 10032], [13325, 13325, 2952, 8645, 10140, 10140], [13325, 13325, 10920, 10373, 8364, 8364], [13481, 11951, 3468, 11440, 4329, 2415], [13585, 10615, 3300, 6519, 8092, 5408], [13585, 12597, 4028, 11088, 5423, 4875], [13728, 9061, 9061, 9020, 9020, 861], [13728, 9061, 9061, 9061, 9061, 1722], [13775, 11687, 8700, 10127, 11248, 6500], [13804, 13260, 10064, 7119, 9065, 9879], [13804, 13520, 1012, 13169, 873, 689], [13920, 7085, 7085, 7076, 7076, 357], [13920, 7085, 7085, 7085, 7085, 714], [13923, 13524, 5397, 11492, 5525, 4240], [13949, 13760, 7293, 12376, 6435, 3432], [13949, 13949, 12870, 13760, 7293, 7293], [13965, 12325, 4972, 3955, 10472, 8580], [14022, 9061, 9061, 7395, 7395, 5236], [14022, 9061, 9061, 9061, 9061, 10472], [14036, 8415, 5863, 6936, 7420, 1887], [14162, 11713, 2595, 2595, 11713, 9220], [14229, 12259, 9100, 10005, 5712, 7084], [14229, 12259, 9100, 12259, 9100, 14168], [14300, 14212, 1584, 13949, 2091, 1365], [14300, 14300, 3168, 13949, 2091, 2091], [14365, 14300, 1365, 14212, 2091, 1584], [14365, 14365, 2730, 14212, 2091, 2091], [14469, 14450, 4541, 761, 14450, 14469], [14637, 10933, 10100, 10100, 10933, 14637], [14664, 11985, 2961, 10360, 9568, 6929], [14703, 13472, 1295, 1972, 12905, 11700], [14820, 10324, 4624, 10101, 6201, 3625], [14861, 9709, 6552, 8723, 6600, 1632], [14875, 13200, 2725, 12852, 4403, 2028], [15080, 14820, 500, 13621, 1557, 1309], [15140, 10556, 9456, 7175, 10485, 6069], [15159, 10441, 7000, 7000, 10441, 6409], [15170, 10175, 7215, 7232, 9282, 3393], [15184, 14841, 1225, 14412, 980, 735], [15400, 13108, 13108, 12155, 12155, 1533], [15540, 12852, 8736, 11315, 9061, 2405], [15555, 14948, 2827, 14820, 4725, 1952], [15600, 9997, 9997, 9605, 9605, 2772], [15600, 9997, 9997, 9997, 9997, 5544], [15677, 15555, 1952, 4725, 14948, 14820], [15677, 15677, 3904, 4725, 14948, 14948], [15677, 15677, 9450, 1952, 15555, 15555], [15677, 15677, 9450, 3904, 15677, 15677], [15725, 15540, 2405, 12852, 9061, 8736], [15725, 15725, 4810, 12852, 9061, 9061], [15840, 14796, 1332, 7975, 7975, 6929], [15900, 15587, 1403, 15312, 4284, 2915], [16031, 15600, 799, 12144, 10465, 9792], [16095, 15776, 2465, 15300, 5865, 4420], [16133, 13108, 9405, 10220, 12483, 8208], [16133, 13108, 9405, 10608, 12155, 7700], [16165, 15900, 2915, 4284, 15587, 15312], [16165, 16165, 5830, 4284, 15587, 15587], [16165, 16165, 8568, 2915, 15900, 15900], [16165, 16165, 8568, 5830, 16165, 16165], [16192, 14645, 1995, 10400, 10400, 8885], [16211, 15725, 1776, 13975, 7740, 8484], end

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Re: [math-fun] (Foucault non)spherical pendulums?
by Bill Gosper 13 Dec '11

13 Dec '11

On Mon, Dec 5, 2011 at 1:44 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > Are they chaotic or just hairy? The Wolfram demonstration seems > quasiperiodic.? > Gene once disabused me of the folly of trying to resolve the motion into x > and y. > But suppose we hung the string from the inside of an upward cusp of a > cycloid of > revolution. Would the pendulum simply describe an ellipse? > --rwg > Gene> While the one-dimensional cycloidic pendulum is isochronous, the two-dimensional one is not. Assume a point mass pendulum. In the one-dimensional case, It is known that in this configuration, the motion lies on a cycloid, whose parametric equation can be taken to be x = a (u + sin u), z = a (1 - cos u). Doing the algebra, and letting s = sin(u/2), the kinetic and potential energies are T = (1/2) m ((dx/dt)^2 + (dz/dt)^2) = 8 m a^2 (ds/dt)^2, V = m g z = 2 m g a s^2. This has the form of a harmonic oscillator, T = (1/2) M (ds/dt)^2, V = (1/2) M ω^2 s^2, and solving for frequency, ω^2 = g/(4 a), verifying the isochronicity since ω is independent of amplitude. Now consider circular motion in the two-dimensional case. The centripetal force in general is m v^2/r = m r (dθ/dt)^2 = m a (u + sin u) (dθ/dt)^2, and for this case is dV/dr = m g (dz/du) / (dr/du) = mg tan(u/2). Then (dθ/dt)^2 = (g/a) tan(u/2) / (u + sin u). Now the frequency increases with amplitude. This makes sense. The mass in an ordinary pendulum moves on a circular arc, but the frequency decreases with amplitude. By bending the circular arc upward into a cycloid, we increase the restoring force, compensate for the decreased frequency, and make the pendulum isochronous. But for circular motion in the two-dimensional case, the pendulum is always in the region of increased force, and so the frequency is higher. -- Gene _______________________________________________ A spherical pendulum that wishes it weren't: http://en.wikipedia.org/wiki/Foucault_pendulum : "A Foucault pendulum requires care to set up because imprecise construction can cause additional veering which masks the terrestrial effect. The initial launch of the pendulum is critical; the traditional way to do this is to use a flame to burn through a thread which temporarily holds the bob in its starting position, thus avoiding unwanted sideways motion. Air resistance <http://en.wikipedia.org/wiki/Air_resistance> damps the oscillation, so some Foucault pendula in museums incorporate an electromagnetic or other drive to keep the bob swinging; ..." Questions: How do you electromagnetically pump it without introducing spurious torques? How do you synchronize with the swinging? How feasible would it be to simply weld the hinge of an ordinary pendulum to the bottom of a freely spinnable rotating shaft? Emphasis on the "freely". --rwg

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[math-fun] Diophantine equation a^4+b^4=d*c^2 and the specialness of 17, 68, 82
by Warren Smith 12 Dec '11

12 Dec '11

>>WDS: Search II: a^4+b^4=d*c^2 with 0<a<b<=1024 and 0<d<=1024. Result: The following d are "special" (allow an apparent infinity of integer solutions, as opposed to none): 17, 68, 82, 97, 113, 153, 193, 257, 272, 274, 328, 337, 388, 425, 433, 452, 514, 577, 593, 612, 626, 641, 673, 706, 738, 772, 833, 873, 881, 914, 1017. [This sequence is not in OEIS.] >Phil Carmody: Whilst the existence of a solution for d = D doesn't guarantee on for d = k^2*d, I'd be tempted to fold all the non-quadratfrei numbers in that list into their core, moving the square part into c. 68 -> 17 153 -> 17 272 -> 17 (* a 4th power, so definitely worth dropping) 328 -> 82 388 -> 97 425 -> 17 452 -> 113 ... --WDS: Actually, the existence of an integer solution (a,b,c) for d=D DOES guarantee the existence of a RATIONAL solution for d=D*k^2, namely use c/k not c. If we then multiply a times k, b times k, and c/k times k^2 we get INTEGERS. So yes, Phil Carmody is entirely justified: we should only consider squarefree d. But the subsequence 17, 82, 97, 113,... of squarefree "special" d also is not in the OEIS. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] polynomials with lots of integer roots computed in few arithmetic operations (Phil Carmody?)
by Warren Smith 12 Dec '11

12 Dec '11

In the book by Crandall & Pomerance, Phil Carmody is mentioned as having found a lot of polynomials of the form Poly(x) = (((x^2-a)^2-b)^2-c)^2-d with all 16 roots being integers. [Is it possible to get 32 integer roots at the next stage?] Might Phil inform us more about how he did that and what are these polynomials? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] Diophantine equation a^4+b^4=d*c^2 and the specialness of 17, 68, 82
by Phil Carmody 12 Dec '11

12 Dec '11

From: Warren Smith <warren.wds(a)gmail.com> > Search II: a^4+b^4=d*c^2 with 0<a<b<=1024 > and 0<d<=1024. > Result: The following d are "special": > 17, 68, 82, 97, 113, 153, 193, 257, 272, 274, 328, 337, 388, 425, 433, > 452, 514, 577, 593, 612, 626, 641, 673, 706, 738, 772, 833, 873, 881, > 914, 1017. > [This sequence is not in OEIS.] Whilst the existence of a solution for d = D doesn't guarantee on for d = k^2*d, I'd be tempted to fold all the non-quadratfrei numbers in that list into their core, moving the square part into c. 68 -> 17 153 -> 17 272 -> 17 (* a 4th power, so definitely worth dropping) 328 -> 82 388 -> 97 425 -> 17 452 -> 113 612 -> 17 738 -> 82 772 -> 193 833 -> 17 873 -> 97 1017 -> 113 I noticed this because hte GP script I wrote split a^4+b^4 into its core and square part, and sorted things by core. Consequently I didn't see any of the above NQF d's. Alas I don't have time to investigate, but I hope a number-theorist wakes up and stokes this rather intersting looking fire. Phil

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