My Favorite Unsolved problem: (4) Rational Box

One of the unsolved problems in mathematics. A number theory question:

Does there exist a rectangular box all of whose edges and diagonals are integers?

A rectangular box is a solid with six rectangular faces.

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Pythagorean Triples

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html

For any odd number a,

[a, (a^2 - 1) / 2, (a^2 + 1) / 2] and
[(a^2 + 1) / 2, (((a^2 + 1)/2)^2 - 1) / 2, (((a^2 + 1)/2)^2 + 1) / 2]

both sets represent Pythagorean triples.

Then,

the diagonal is .... (((a^2 + 1) / 2)^2 + 1) / 2

getting the equation....

a^2 + ((a^2 - 1) / 2)^2 + ((((a^2 + 1) / 2)^2 - 1) / 2)^2 = ((((a^2 + 1) / 2)^2 + 1) / 2)^2

Can "a" be an integer?

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Related topics:

Euler Brick

http://mathworld.wolfram.com/EulerBrick.html

Perfect Cuboid

http://mathworld.wolfram.com/PerfectCuboid.html

Heronian Tetrahedron

http://mathworld.wolfram.com/HeronianTetrahedron.html

Math in the News

The Mathematics of Terrorism

Seemingly random attacks contain an unexpected regularity: the same numerical pattern seen in Wall Street booms and busts.

http://discovermagazine.com/2010/jul-aug/07-the-mathematics-of-terrorism

The Math Behind Beauty

http://discovermagazine.com/2007/jun/blinded-by-science

Computer scientists make progress on math puzzle

http://www.physorg.com/news/2010-10-scientists-math-puzzle.html

The friendly way to catch the flu: 'Friendship paradox' may help predict spread of infectious disease

http://www.physorg.com/news203782450.html

Using mathematics to identify the good guys

http://www.physorg.com/news/2010-10-mathematics-good-guys.html

Dealing with the unexpected

To regain balance from an unexpected slip on the ice can require an abundance of rapid movement, but conscious thought isn't part of the equation.

http://www.physorg.com/news/2010-10-unexpected.html

Microorganisms offer lessons for gamblers and the rest of us

When it comes to gambling, many people rely on game theory, a branch of applied mathematics that attempts to measure the choices of others to inform their own decisions. It's used in economics, politics, medicine -- and, of course, Las Vegas. But recent findings from a Tel Aviv University researcher suggest that we may put ourselves on the winning side if we look to bacteria instead.

http://www.physorg.com/news/2010-10-microorganisms-lessons-gamblers-rest.html

Women's choices, not abilities, keep them out of math-intensive fields

http://www.physorg.com/news/2010-10-women-choices-abilities-math-intensive-fields.html

An Exhibition That Gets to the (Square) Root of Sumerian Math

http://www.nytimes.com/2010/11/23/science/23babylon.html?_r=1

My Favorite Unsolved problem: (3) Egyptian Fraction

A fraction written as a sum of distinct unit fractions is called an Egyptian Fraction.

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html

E.g.

3/4 = 1/2 + 1/4

6/7 = 1/2 + 1/3 + 1/42

5/8 = 1/2 + 1/8

If n is an integer larger than 1, must there be integers x, y, and z, such that

4/n = 1/x + 1/y + 1/z ?

A number of the form 1/x where x is an integer is called an Egyptian fraction.

Thus, we want to know if 4/n is always the sum of three Egyptian fractions, for n > 1.

..............................................................

If 4/n = 1/x + 1/y + 1/z,

then there exists a third degree polynomial x^3 + ax^2 + bx + c

where 4/n = b/c for integers b and c.

This is because 1/x + 1/y + 1/z = (z(x + y) + xy)/xyz.

And the polynomial

(a + x)(a + y)(a + z) = a^3 + (x + y + z)a^2 + (z(x + y) + xy)a + xyz,

which proves the statement when b = z(x + y) + xy and c = xyz.




......................................................

Here's a much easier problem:

(1) 1/a + 1/b + 1/c + 1/d + 1/e + 1/f = 1

(2) 1/a + 1/b + 1/c + 1/d + 1/e + 1/f + 1/g = 1

(3) 1/a + 1/b + 1/c + 1/d + 1/e + 1/f + 1/g + 1/h = 1

Find a, b, c, d, e, f, g, h

They are all positive, different, whole numbers.

My Favorite Unsolved problem: (2) Goldbach's conjecture

Goldbach's conjecture is one of the oldest unsolved problems in Number theory and in all of mathematics.

It states:

Every even integer greater than 2 is a Goldbach number, a number that can be expressed as the sum of two primes.

E.g. 50 = 3 + 47

http://en.wikipedia.org/wiki/Goldbach's_conjecture

Fun with num3ers: My Favorite Unsolved problem: (1) Hailstone Num3er...

Fun with num3ers: My Favorite Unsolved problem: (1) Hailstone Num3er...: "Starting with any positive integer n, form a sequence in the following way>> If n is even, divide it by 2 to give n' = n/2.>> If n is odd, m..."

My Favorite Unsolved problem: (1) Hailstone Num3ers

I have a short list of favorite unsolved problems in Number Theory. I'll start with the Hailstone numbers, aka the Hailstone sequences:

Starting with any positive integer n, form a sequence in the following way

>> If n is even, divide it by 2 to give n' = n/2.

>> If n is odd, multiply it by 3 and add 1 to give n' = 3n + 1.

E.g.

n = 5 ---> 16, 8, 4, 2, 1, 4, 2, 1, ...

n = 11 --> 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ...

n = 13 --> 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ...

n = 27 --> 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ...

n = 100 ---> 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1,...

Can it be proved that every starting value will generate a sequence that eventually settles to 4, 2, 1, 4, 2, 1,...?

Try and generate more sequences with the Hailstone Evaluator

http://plus.maths.org/content/mathematical-mysteries-hailstone-sequences

This is also known as the Collatz Problem

http://mathworld.wolfram.com/CollatzProblem.html

It works for every number you try. Does it mean that it would work for all numbers?

No one knows for sure. Just because it works for every number you try -- big or small -- doesn't guarantee that it works for all numbers.

And, why the pattern (4, 2, 1, 4, 2, 1,) keeps popping up?

Numbers on car number plates

I've been wondering about the numbers on car license plates: some are 3-digit, others are 4-digit or 5-digit numbers.

United States and Canada

http://en.wikipedia.org/wiki/Vehicle_registration_plate#United_States_and_Canada

In the picture:

Saskatchewan ----------> 546

Michigan ------------------> 6610

Brunswick ----------------> 693

Northwest Territories --> 38542

If we separate them into groups, such as, Group A: the 3-digits license plates, Group B: the 4-digits, Group C: the 5-digits, then,

If we multiply the 3-digits on car number plates together what number are we most likely to get?

Then, if we do do that with the 4-digits, then with the 5-digits.

In this very small sample, we get:

Saskatchewan ............. 5 * 4 * 6 = 120

Brunswick ................... 6 * 9 * 3 = 162

Michigan ..................... 6 * 6 * 1 * 0 = 0

Northwest Territories .... 3 * 8 * 5 * 4 * 2 = 960

I don't have the necessary info to continue.

Somewhere, out there, someone may be able to.

3-digit numbers

Write any three-digit number whose first digit differs from its last by 1.

E.g. 314, 647, 738, etc.

Now, reverse the digits then subtract the smaller from the larger number:

413 - 314 = 99

746 - 647 = 99

837 - 738 = 99

Does this always happen?

Square / triangular / pentagonal num3ers

Square number

http://en.wikipedia.org/wiki/Square_number

Triangular number

http://en.wikipedia.org/wiki/Triangular_number

Triangular numbers: a(n) = C(n+1,2) = n(n+1)/2 = 0+1+2+...+n.

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, ...

A SQUANGULAR number is a number that is simultaneously triangular and square.

The first few are : 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025, ...

http://en.wikipedia.org/wiki/Square_triangular_number

Pentagonal number

http://en.wikipedia.org/wiki/Pentagonal_number

Pentagonal numbers: n(3n-1)/2 :

0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, ...

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Question: Can you find a number that is simultaneously triangular, square and pentagonal?

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3-digit / 4-digit palindromes / Primes

Question #1: to find the number of palindromic and prime numbers that can appear on a 24-hour and/or 12-hour digital clock.

==============================================================

I'll list all the 3-digit/4-digit and prime numbers (not all of them will apply.)

3-digit palindromes

101, 111, 121, 131, 141, 151, 161, 171, 181, 191

202, 212, 222, 232, 242, 252, 262, 272, 282, 292

303, 313, 323, 333, 343, 353, 363, 373, 383, 393

404, 414, 424, 434, 444, 454, 464, 474, 484, 494

505, 515, 525, 535, 545, 555, 565, 575, 585, 595

606, 616, 626, 636, 646, 656, 666, 676, 686, 696

707, 717, 727, 737, 747, 757, 767, 777, 787, 797

808, 818, 828, 838, 848, 858, 868, 878, 888, 898

909, 919, 929, 939, 949, 959, 969, 979, 989, 999

4-digit palindromes

1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991

2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992

3003, 3113, 3223, 3333, 3443, 3553, 3663, 3773, 3883, 3993

4004, 4114, 4224, 4334, 4444, 4554, 4664, 4774, 4884, 4994

5005, 5115, 5225, 5335, 5445, 5555, 5665, 5775, 5885, 5995

6006, 6116, 6226, 6336, 6446, 6556, 6666, 6776, 6886, 6996

7007, 7117, 7227, 7337, 7447, 7557, 7667, 7777, 7887, 7997

8008, 8118, 8228, 8338, 8448, 8558, 8668, 8778, 8888, 8998

9009, 9119, 9229, 9339, 9449, 9559, 9669, 9779, 9889, 9999

There are 90 palindromic numbers with three digits, and 90 palindromic numbers with four digits.

And now, the 3-digit Primes :

101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167,

173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241,

251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331,

337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419,

421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499,

503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599,

601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677,

683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773,

787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877,

881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977,

983, 991, 997,

The 4-digit Primes :

1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093,

1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213,

1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303,

1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439,

1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543,

1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627,

1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753,

1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877,

1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999,

2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111,

2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239,

2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347,

2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447,

2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593,

2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699,

2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801,

2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927,

2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061,

3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203,

3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323,

3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457,

3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557,

3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673,

3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797,

3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919,

3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049,

4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159,

4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283,

4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441,

4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561,

4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679,

4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813,

4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957,

4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077,

5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209,

5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351,

5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477,

5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591,

5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717,

5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849,

5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987,

6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121,

6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257,

6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359,

6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521,

6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659,

6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781,

6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907,

6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019,

7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187,

7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321,

7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487,

7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583,

7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703,

7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867,

7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009,

8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147,

8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273,

8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423,

8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573,

8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681, 8689, 8693,

8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819,

8821, 8831, 8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, 8933, 8941, 8951,

8963, 8969, 8971, 8999, 9001, 9007, 9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091,

9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221,

9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349,

9371, 9377, 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467,

9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623,

9629, 9631, 9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749,

9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871,

9883, 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973

Buying and Selling

Imagine that you bought an item for $6, sold it for $7, bought it back for $8, then sold it for $9.

How much profit did you make?

Buying and Selling

Imagine that you bought an item for $6, sold it for $7, bought it back for $8, then sold it for $9.

How much profit did you make?

Benoît Mandelbrot

http://en.wikipedia.org/wiki/Beno%C3%AEt_Mandelbrot

There are many quotes by Benoît Mandelbrot, but there's one quote I like the most:

"One hope for regularity but one lives in roughness."

Benoît Mandelbrot did critical work in topics such as fat tails and fractals.

I'll go back and read : "The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward"

http://www.amazon.com/Misbehavior-Markets-Fractal-Financial-Turbulence/dp/0465043577

A Panorama of Fractals and Their Uses :

http://classes.yale.edu/fractals/panorama/welcome.html

Benoit Mandelbrot: Fractals and the art of roughness

Can a number "contain itself"?

Does pi contain pi?

No. If pi contained pi, at the point where the "contained pi" started, pi would become a repeating decimal. Then pi would not be irrational. Hence, pi cannot contain pi.

Is there any way that a number can "contain itself" other than it having a pattern that repeats itself?

Here's another way to look at it.

If a number contained itself starting after n digits, then the first n digits would be the same as the second n digits. Similarly, the second n digits of the copy would be the same as the first n digits of the copy which means, since the second n digits of the copy are the third n digits of the original number, the third n digits of the original number would be the same as the first n digits. And so on, and so on, to infinity. The number would be a repeating decimal of period n.

Fun with num3ers: Consecutive numbers

Fun with num3ers: Consecutive numbers: "Between 1000 and 2000 you can get each integer as the sum of nonnegative consecutive integers. For example, 147 + 148 + 149 + 150 + 151 + 1..."

Consecutive numbers

Between 1000 and 2000 you can get each integer as the sum of nonnegative consecutive integers.

For example,

147 + 148 + 149 + 150 + 151 + 152 + 153 = 1050

There is only one number that you cannot get.

What is this number?

Answer : it's the number 1024.

Could you explain why?

Fun with num3ers: Four 3-digit Numbers

Fun with num3ers: Four 3-digit Numbers: "There are four 3-digit natural numbers, each of them equals the sum of the cubes of its digits. Three of them are: 153 = 1^3 + 5^3 + 3^3 = 1..."

Four 3-digit Numbers

There are four 3-digit natural numbers, each of them equals the sum of the cubes of its digits.

Three of them are:

153 = 1^3 + 5^3 + 3^3 = 1 + 125 + 27

370 = 3^3 + 7^3 + 0^3 = 27 + 343 + 0

407 = 4^3 + 0^3 + 7^3 = 64 + 0 + 343

Do you know what the fourth one is?

It's 371.

371 = 3^3 + 7^3 + 1^3 = 27 + 343 + 1

Recreation math & fun with Num3ers.

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Semiprime: a semiprime is a number of the form p*q where p and q are primes, not necessarily distinct.

http://en.wikipedia.org/wiki/Semiprime

http://mathworld.wolfram.com/Semiprime.html

For example,

842 (= 2 * 421) is a semiprime sandwiched between semiprimes: 841 and 843

841 = 29 * 29 = 29^2

843 = 3 * 281

Here's the list of the first 10,000 Primes :

http://primes.utm.edu/lists/small/10000.txt

For example, between consecutive primes 113 and 127 there are six semiprimes:

115 (=5*23), 118 (=2*59), 119 (=7*17), 121 (=11*11), 122 (=2*61), 123 (=3*41).

Could you find others?

And find a prime gap that is less and more than six semiprimes.