Erdos Discrepancy Problem

John reminded that a native of South Australia, Terence Tao, has only last year finally solved the Erdos Discrepancy Problem. Here is an excerpt from Wikipedia about the problem:

Here is a link that explains it in less mathematical terms.

Anti-divisors

I'd never heard of an anti-divisor until I reached my 24434th day on Earth. Here is my tweet for that day:

24434: 2×19×643; member of OEIS A258786: numbers \(n\) whose sum of anti-divisors is a permutation of their digits 

Of course, I needed to find out exactly what an anti-divisor was. Fortunately, I came across a subsite on OEIS that explained the concept. Here is the explanation:

The anti-divisor, or unbiased non-divisor, is a concept related very closely to the concept of prime numbers, and the concept of a divisor. 

Every integer is either prime, or has two or more prime factors, for example 61 is prime, but 63 can be written as 3 x 3 x 7. 

Every integer is then said to be the product of some factors. A divisor is a combination of these factors, for example 63 has the divisors 1, 3, 7, 9, 21 and 63, as 63 divided by any of these numbers yields an integer. 

It logically follows from this that any number that is not a divisor of an integer is a non-divisor. So, we say that the non-divisors of 63 are all the integers less than or equal to 63 except for 1, 3, 7, 9, 21 and 63. 

The definition of an anti-divisor follows on from this - an anti-divisor is a non-divisor such that doesn't divide the number in the most unbiased way possible. For example, we say 42 is an anti-divisor of 63, as 42 surrounds 63 with a gap of 21 on either side. 41 on the other hand, has gaps of 22 and 18, so the gap of 22 is larger than the 18. 41 is called a biased non-divisor of 63. 

There are two distinct mathematical definitions of anti-divisors, one for even anti-divisors, and one for odd anti-divisors. The two definitions are very similar, however we need two definitions because a even anti-divisor candidate defines exactly one number, and an odd anti-divisor candidate defines exactly two numbers. 

Both definitions do share a common feature. An anti-divisor of n must lie in the region [2, \(n\)-1]. 1 is never an anti-divisor, its exclusion is determined by the phrase 'non-divisor', and the fact that no integers are contained between two successive integers with 1 as a divisor. Also note that there are 'larger-than '\(n\)' anti-divisors, namely 2\(n\)-1, 2\(n\) and 2\(n\)+1, but these are considered to be trivial. 

For k even:

\(k\) is an even anti-divisor of \(n\) when \(n \equiv k/2 \bmod{k} \) or to put it another way, \(k\) is an even anti-divisor of \(n\) when \(k(x+1/2)=n\), for some \(x\) greater than or equal to 1. For example 10 is an even anti-divisor for 15, 25, 35, 45, 55, etc,.. and no other numbers. This is because \(10(x+1/2)=10x+5\), and so the numbers are generated over \(x>1\). Another example, 8 is an even anti-divisor for 12, 20, 28, etc..., and no other numbers. 

For k odd: 

\(k(x+1/2)\) falls between 2 integers, and we claim that both of these integers are anti-divisors for \(n\).

Hence the basic definition is the same for odd anti-divisors, except that \( k(x+1/2) \) itself is never an integer, so we define the odd anti-divisors as the two numbers surrounding \(k(x+1/2) \). So \(k\) is an odd anti-divisor of \(n\) when we have either \(k(x+1/2)-1/2\) or \(k(x+1/2)+1/2\) for some \(x\) greater than or equal to 1.  

In mod terminology, \(k\) is an odd anti-divisor of \(n\) when \(n \equiv (k-1)/2 \bmod{k} \) or \( n \equiv (k+1)/2 \bmod{k} \). So, 7 is an odd anti-divisor for 10, 11, 17, 18, 23, 24, etc.., and no other numbers. This is because we have the two generating functions \(7x+3\) and \(7x+4\). Again, 11 is an odd anti-divisor for \(11x+5\), \(11x+6\), i.e. 16, 17, 27, 28, etc..., and no other numbers. 

Putting it together 

Now we have defined an anti-divisor, let's consider an integer n, e.g.10. We then find all the anti-divisors of 10, in this case 3, 4 and 7. The pattern of anti-divisors is as random and incomprehensible as that with prime numbers. 

See another post from 28th February 2021 titled More on Anti-divisors. 

Gaps between Twin Primes

Today is a prime day for me, the prime being the larger half of the twin prime pair 24419 and 24421. Just as the gaps between primes are variable, so too are the gaps between successive twin primes. However, at certain points records are set regarding how big these gaps are. It just so happens that 24421 marks one of those points. The next prime pair is 24917 and 24919, and the gap of 496 between 24421 and 24917 sets a record. A list of the initial record intervals is attached.

24421 is a member of OEIS A036061: increasing gaps among twin primes, the largest prime of the starting twin pair. The record gaps between primes was treated in this earlier post.

Largest Prime

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 \(2^{11} - 1 = 2047 = 23 \times 89\) is not prime. Here are links to some more interesting information about Mersenne primes:

• Source 1
• Source 2

on 27th of October 2024

Update: \(2^{136,279,841} - 1\) has \(41,024,320\) digits and is prime! Read all about the new largest prime number ever found: Stand-up MathsYouTube video.

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.

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.

Double and Reverse Digits

After twelve days, I encountered today the first member of the twin prime pair: 24371 and 24373. It's been a while: the last pair was 24179 and 24181 as far as I can tell. The number is a member of the interesting OEIS A036447 formed using 1 as its starting point and then doubling and reversing the digits:

1, 2, 4, 8, 61, 221, 244, 884, 8671, 24371, ...

The number is also a member of OEIS A243408: primes p such that 10p-1, 10p-3, 10p-7 and 10p-9 are all prime. This means that 243709, 243707, 243703 and 243701 are all prime.

Additionally, the number is a member of OEIS A158641: strong primes p: adding 2 to any one digit of p produces a prime number (no digits 8 & 9 in p). This means that 44371, 26371, 24571, 24391 and 24373 are prime.

There's still more. The number is a member of OEIS A104846: primes from merging of 5 successive digits in decimal expansion of e. Here is part of the sequence (up to 24371):

74713, 62497, 24977, 24709, 47093, 95957, 49669, 27427, 46639, 32003, 59921, 21817, 35729, 63073, 28627, 27943, 94349, 33829, 98807, 57383, 41879, 18793, 91499, 68477, 47741, 37423, 42437, 24371

Lastly, 24371 is also a member of OEIS A054564 as describe below: