If there might be interesting for you summary knowledge about numbers called Münchhausen numbers up to now, read my new paper. Here.
Search For Numbers
I assume that my puzzles will mainly be focused on machine use / break. Sure, writing good code is a commendable job. However, more mathematical procedures, such as proofs, are welcome, too. Thank you for your interest and feedbacks. Miroslav Kureš
čtvrtek 1. dubna 2021
neděle 21. června 2020
Puzzle 3: Münchhausen numbers of length 3
The number 3435 = 33+44+33+55 is well-known as the Münchhausen number in the base 10 and its length (number of digits) is 4.
More examples:
The number 96446 is the Münchhasuen number in the base 9 and its length is 6, because it has 6 digits in the base 9:
9644610 = 1562629
digits 1, 5, 6, 2, 6, 2
96446 = 11+55+66+22+66+22
The number 20017650854 is the Münchhasuen number in the base 12 and its length is 10, because it has 10 digits in the base 12:
2001765085410 = 3a67a5483212
digits 3, a, 6, 7, a, 5, 4, 8, 3, 2
20017650854 = 33+1010+66+77+1010+55+44+88+33+22 (the digit a is ten)
Bases can be considered arbitrary. Then it is not difficult to prove that there are infinitely many Münchhasuen numbers of length 2 in some base. In particular, 1+nn is of length 2 in the base 1+nn-n for each n greater than 1.
But what about length 3? Are there infinitely many Münchhausen numbers of length 3? Up to now, we know only these Münchhasuen numbers of length 3:
29 in base 4
55 in base 4
3153 in base 25
49782 in base 91
46661 in base 215
New: 823545 in base 904 (June 21, 2020; by mersenneforum.org user axn)
New: 823545 in base 904 (June 21, 2020; by mersenneforum.org user axn)
sobota 28. prosince 2019
Puzzle 2: Happy New Year 2020 (and the link with 2- 4- 6- concecutive prime gaps)
Gaps between prime numbers represent an interesting and widely
studied topic. Taking the first few primes
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
there are gaps between them
1, 2, 2, 4, 2, 4, 6, ...
We see that the smallest gap 1 is an exception: regardless
of this, gaps are always positive even numbers. The gap 2 is between so-called
prime twins, the smallest such twins are 3 and 5. The prime twins may be
infinitely many, but proof of this has not yet been given.
The largest gap does not exist because a gap can be of any
length. This follows from this finding: if we take any natural number n,
the numbers n! +2, n! +3, ..., n! + n are all
composed (surely, n! +2 is divisible by two, n! +3 is divisible
by three, and so on) and is therefore a part of some gap between prime
numbers.
As already mentioned, primes 3, 5 are smallest consecutive
primes with gap 2. Furthermore:
· Prime numbers 5, 7, 11 are the smallest consecutive primes
with gaps 2, 4.
· Prime numbers 17, 19, 23, 29 are the smallest consecutive
prime numbers with gaps 2, 4, 6.
· Prime numbers 347, 349, 353, 359, 367 are the smallest
consecutive prime numbers with gaps 2, 4, 6, 8.
· Prime numbers 13901, 13903, 13907, 13913, 13921, 13931 are
the smallest consecutive prime numbers with gaps 2, 4, 6, 8, 10.
In this way, we could continue up to a string of fifteen
consecutive prime numbers, each with by two increasing gaps from 2 to 28, which
is the last result of this kind. The first prime
number here is 221860944705726407. Finding the following string of 16 consecutive primes with gaps from 2 to 30 can
be a challenge for the reader (Puzzle 2). See OEIS
A016045.
Let us look at four consecutive prime numbers 17, 19, 23,
29, which are the smallest with gaps 2, 4 and 6. If we calculate the sum of
squares
172
+ 192 + 232 + 292,
the result is 2020. Pour
féliciter 2020.
úterý 9. července 2019
Puzzle 1: Sum of Powers of Primes / The paper
Now, I have written known facts about numbers in question in
my paper published in the journal Notes on Number
Theory and Discrete Mathematics; not only that, I tried to derive some
properties of these numbers. I'll be glad for your eventual response to this paper.
The link to it is here.
sobota 15. prosince 2018
2019
Sure, the number 2019 has a lot of great properties. For
example, 2019 is the least number having
six different representations as a sum of three squares of prime numbers.
2019 = 232+232+312
2019 = 172+192+372
2019 = 112+232+372
2019 = 132+132+412
2019 = 72+172+412
2019 = 72+112+432
There
a lot of numbers having six different representations as above, not seven or
more. They form a sequence 2019, 2091, 2499, 3099, 3219, 3339, 3579,
3939, 4011, ... E.g., 4011 = 312 + 372 +
412 = 292 + 312 + 472 =
112 + 412 + 472 = 192 +
292 + 532 = 132 + 192 +
592 = 112 + 132 + 612.
čtvrtek 6. září 2018
Puzzle 1: Sum of Powers of Primes / Results - 2
Dana
Jacobsen has published a short code for (P2) in the Perl language on mersenneforum. He notes that testing
up to 109 takes 90 minutes.
středa 5. září 2018
Puzzle 1: Sum of Powers of Primes / Results - 1
The mersenneforum
user, nicknamed a1call, has
announced the finding of a new number with the (P2) property. It is the number 9634877 (Sep 3rd, 2018). Excellent work!
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