The news of the discovery of a new, largest known prime broke about a week ago but I've only gotten around to writing about it here. It was of course a Mersenne prime discovered via GIMPS, the Great Internet Mersenne Prime Search. The number containing 22,338,618 digits is 2^74,207,281 - 1 where 74,207,281 itself must be prime of course. It is the 49th known Mersenne prime defined as a prime expressible in the form 2^p - 1 where p is prime. The first Mersenne primes are 3, 7, 31, and 127 corresponding to p values of 2, 3, 5, and 7 respectively. Note that p being prime is not sufficient to ensure that 2^p - 1 will be prime. As a counter example take p=11. The resulting number 2^11 - 1 = 2047 = 23 x 89 is not prime. Here are links to some more interesting information about Mersenne primes:
Wednesday 27 January 2016
Monday 11 January 2016
Cubic Numbers
Today, January 11th 2016, I'm 24389 days old and what's special is that this number is 29 cubed or 29 x 29 x 29. Days like this are rare. For example, \(28^3 \) or 21592 occurred on May 10th 2009 and \(30^3 \) or 27000 will occur on March 6th 2023. Cubes of prime numbers are even rarer of course. The prime preceding 29 is 23 and \(23^3 \) or 12167 occurred on July 26th 1982. The prime following 29 is 31 and \( 31^3 \) or 29791 will occur on October 26th 2030 when I'm 81 years of age (if I make it that far).
24389 has a surprisingly large number of entries in the Online Encyclopaedia of Integer Sequences (OEIS), 174 in fact which is unusual for a composite number of this magnitude. The first entry is for OEIS A000578: the cubes \( a(n) = n^3 \). The sequence, up to 24389 when n=29, looks like this:
0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389
The next entry is OEIS A030078: cubes of primes. The sequence, up to 24389, is: 8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389.
From WolframAlpha, we find that 24389 is also a cube that is expressible as the sum of two squares in two different ways:
\(24389 = 58^2+145^2 = 65^2+142^2 \)
Additionally, we find that 24389 is the hypotenuse of a primitive Pythagorean triple:
\(24389^2 = 15939^2+18460^2 \). So, all in all, an interesting number.
24389 has a surprisingly large number of entries in the Online Encyclopaedia of Integer Sequences (OEIS), 174 in fact which is unusual for a composite number of this magnitude. The first entry is for OEIS A000578: the cubes \( a(n) = n^3 \). The sequence, up to 24389 when n=29, looks like this:
0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389
The next entry is OEIS A030078: cubes of primes. The sequence, up to 24389, is: 8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389.
From WolframAlpha, we find that 24389 is also a cube that is expressible as the sum of two squares in two different ways:
\(24389 = 58^2+145^2 = 65^2+142^2 \)
Additionally, we find that 24389 is the hypotenuse of a primitive Pythagorean triple:
\(24389^2 = 15939^2+18460^2 \). So, all in all, an interesting number.
Thursday 7 January 2016
22, Reverse and Add
24384 is a member of OEIS A061561: Trajectory of 22 under the Reverse and Add! operation carried out in base 2. The terms of the sequence, up to and including 24384, are 22, 35, 84, 105, 180, 225, 360, 405, 744, 837, 1488, 1581, 3024, 3213, 6048, 6237, 12192, 12573, 24384. Even though the operations are carried out in base 2, the numbers of this sequence are shown in denary form. The actual base 2 sequence (OEIS A058042: Trajectory of binary number 10110 under the operation 'Reverse and Add!' carried out in base 2) looks like this:
10110, 100011, 1010100, 1101001, 10110100, 11100001, 101101000, 110010101, 1011101000, 1101000101, 10111010000, 11000101101, 101111010000, 110010001101, 1011110100000, 1100001011101, 10111110100000 and on and on it goes ...
22 or 10110 is chosen as the first term because it is the smallest number whose base 2 trajectory does not contain a palindrome. So starting with 10110, the reverse is 01101 and 10110 + 11101 = 100011 and so it goes.
The equivalent sequence in base 10 starts with 196 because, according to this comment for OEIS A006960, 196 is conjectured to be the smallest initial term which does not lead to a palindrome. John Walker, Tim Irvin and others have extended the trajectory of 196 to millions of digits without finding a palindrome.
The Reverse and Add! sequence starting with 196 looks like this:
196, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, 18211171, 35322452, 60744805, 111589511, 227574622, 454050344, 897100798, 1794102596, 8746117567, 16403234045, 70446464506, 130992928913, 450822227944, 900544455998, 1800098901007 and on and on it goes ...
ADDENDUM (added 1st June 2019):
Most numbers do become palindromes fairly quickly under the reverse and add algorithm. OEIS A023109 shows the smallest number that requires exactly \(n\) iterations of Reverse and Add to reach a palindrome. The initial terms, up to \(n=55\) and starting with \(n=0\) are:
0, 10, 19, 59, 69, 166, 79, 188, 193, 1397, 829, 167, 2069, 1797, 849, 177, 1496, 739, 1798, 10777, 6999, 1297, 869, 187, 89, 10797, 10853, 10921, 10971, 13297, 10548, 13293, 17793, 20889, 700269, 106977, 108933, 80359, 13697, 10794, 15891, 1009227, 1007619, 1009246, 1008628, 600259, 131996, 70759, 1007377, 1001699, 600279, 141996, 70269, 10677, 10833, 10911
More information at this later blog post.
10110, 100011, 1010100, 1101001, 10110100, 11100001, 101101000, 110010101, 1011101000, 1101000101, 10111010000, 11000101101, 101111010000, 110010001101, 1011110100000, 1100001011101, 10111110100000 and on and on it goes ...
22 or 10110 is chosen as the first term because it is the smallest number whose base 2 trajectory does not contain a palindrome. So starting with 10110, the reverse is 01101 and 10110 + 11101 = 100011 and so it goes.
The equivalent sequence in base 10 starts with 196 because, according to this comment for OEIS A006960, 196 is conjectured to be the smallest initial term which does not lead to a palindrome. John Walker, Tim Irvin and others have extended the trajectory of 196 to millions of digits without finding a palindrome.
The Reverse and Add! sequence starting with 196 looks like this:
196, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, 18211171, 35322452, 60744805, 111589511, 227574622, 454050344, 897100798, 1794102596, 8746117567, 16403234045, 70446464506, 130992928913, 450822227944, 900544455998, 1800098901007 and on and on it goes ...
ADDENDUM (added 1st June 2019):
Most numbers do become palindromes fairly quickly under the reverse and add algorithm. OEIS A023109 shows the smallest number that requires exactly \(n\) iterations of Reverse and Add to reach a palindrome. The initial terms, up to \(n=55\) and starting with \(n=0\) are:
0, 10, 19, 59, 69, 166, 79, 188, 193, 1397, 829, 167, 2069, 1797, 849, 177, 1496, 739, 1798, 10777, 6999, 1297, 869, 187, 89, 10797, 10853, 10921, 10971, 13297, 10548, 13293, 17793, 20889, 700269, 106977, 108933, 80359, 13697, 10794, 15891, 1009227, 1007619, 1009246, 1008628, 600259, 131996, 70759, 1007377, 1001699, 600279, 141996, 70269, 10677, 10833, 10911
More information at this later blog post.
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