Browsed by
Tag: numbers

Fibonacci Rondo (Rondo #1), Op. 54 (2008) – Musical Composition and Video by G. Stolyarov II

Fibonacci Rondo (Rondo #1), Op. 54 (2008) – Musical Composition and Video by G. Stolyarov II

Comments 0 Comment
The New Renaissance Hat
G. Stolyarov II
November 7, 2014
******************************

The Fibonacci Rondo, a 2008 composition by Mr. Stolyarov, was inspired by the Fibonacci Sequence of numbers, where each subsequent number is the sum of the two previous numbers. If the Fibonacci Sequence begins with 1 and 1, then the first six numbers of the sequence are 1, 1, 2, 3, 5, and 8.

The recurring theme of this composition – which occurs once at 0:32 and again at 1:30 represents musically the beginning of the Fibonacci Sequence and the process of its formation.

If we assign the value 1 to the note C, then we can assign the following values to other notes in relation to it:

2 = D

3 = E

5 = G

8 = C one octave above the “1” note.

Then, through two eighth notes, we can represent the numbers being added, while the following quarter note represents their result.

So two eighth-note C’s will be followed by a quarter-note D to represent “1 + 1 = 2.”

Then the eighth notes C and D, followed by a quarter-note E, represent “1 + 2 = 3.”

Then the eighth notes D and E, followed by a quarter-note G, represent “2 + 3 = 5.”

Then the eighth notes E and G, followed by a quarter-note C from the next octave, represent “3 + 5 = 8.”

Thereafter, the same pattern is applied to other harmonies – both major and minor – to ensure a melodic progression.

The timpani accompaniment in the second appearance of the theme relates this basic structure without any other notes added to reinforce the harmony. Quite a bit of harmonic reinforcement is added in the parts for all the other instruments, however.

This composition is written for a piano, two string sections, and timpani, and remastered using the Finale 2011 software. It probably could not be played by a human orchestra, as the 32nd notes in one of the string sections are simply too fast to be played by human musicians. The ability to reproduce music of this sort is yet another way in which computers have expanded the range of human creativity.

Download the MP3 file of this composition here.

See the index of Mr. Stolyarov’s compositions, all available for free download, here.

Remember to LIKE, FAVORITE, and SHARE this video in order to spread rational high culture to others.

ECM Distributed Computing Project and Mr. Stolyarov Discover Factor for 279-Digit Number (“9” Surrounded by 139 Instances of “7” Per Side)

ECM Distributed Computing Project and Mr. Stolyarov Discover Factor for 279-Digit Number (“9” Surrounded by 139 Instances of “7” Per Side)

Comments 1 comment
The New Renaissance Hat
G. Stolyarov II
November 21, 2012
******************************

I am pleased to report that a large prime factor for (7·10279+18·10139-7)/9 (visualized as a “9” surrounded by 139 instances of “7” on each side – for a total of  279 digits) was discovered on November 19, 2012, through my participation in the ECM distributed computing project (organized via Yoyo@home). This is the fourth discovery made on my computer via the ECM project (see posts about previous discoveries here and here).

The prime factor is a 53-digit number: 42684752427275029312252733896207947190538122452468697. I am credited with the discovery here and here.

I continue to be impressed by the potential of individual hyper-empowerment through distributed computing, and I encourage my readers to also donate their idle computer time to projects that attract their interest.

Factors for (141^141 + 142^142) and (148^148 + 149^149) Discovered by Mr. Stolyarov and ECM Distributed Computing Project

Factors for (141^141 + 142^142) and (148^148 + 149^149) Discovered by Mr. Stolyarov and ECM Distributed Computing Project

Comments 2 comments
The New Renaissance Hat
G. Stolyarov II
August 28, 2012
******************************

Since the discovery in April 2012 of a factor for (118^67 + 67^118), I have continued to donate extensive computing resources to the ECM distributed computing project (organized via Yoyo@home). Today I am pleased to announce that two more large factors of even larger numbers have been discovered as a result of this endeavor. I am credited with the discoveries here. The following is now known:

● The number (141^141 + 142^142) (see long form here) has a 33-digit factor: 168,853,190,844,095,597,109,245,277,698,729.

● The number (148^148 + 149^149) (see long form here) has a 28-digit factor: 9,055,497,748,306,357,299,810,062,467.

To date, my computer has examined 2729 project workunits (each involving an attempt to factor a large number). I have thus far accumulated 528,533.42 BOINC credits for the ECM project.

The magnitudes involved are astounding, considering that the factors discovered are several hundred orders of magnitude less than the original numbers. As an example, (148^148 + 149^149) is equal to approximately 6.39 * 10^323. And yet our advancing technology is enabling us already to explore these immense quantities and derive meaningful conclusions regarding them.

ECM Distributed Computing Project and Mr. Stolyarov Discover Factor for 118^67 + 67^118

ECM Distributed Computing Project and Mr. Stolyarov Discover Factor for 118^67 + 67^118

Comments 0 Comment
The New Renaissance Hat
G. Stolyarov II
April 6, 2012
******************************

I am pleased to announce that my participation in the ECM distributed computing project (organized via Yoyo@home) has resulted in the discovery of a hitherto unknown factor for a large number. I am credited with the discovery here.

Did you know that the number 118^67 + 67^118 (see its long form here) has a multiplicative factor of 2091937057168244837833711997693707725557784572281 – a formidable 49-digit number?

Well, now you know, because of all the computing power I have devoted to the ECM project since late 2011. Finding such large factors of even larger numbers is a rarity. My computer had to examine 1546 project workunits (each involving an attempt to factor a large number) before finding one that resulted in a new discovery. I have thus far accumulated 277,345.88 BOINC credits for the ECM project.

ECM is a free distributed computing project that anyone can participate in. Its goal is to find factors for large numbers using the method of Elliptic Curve Factorization. It is highly rewarding to be able to devote otherwise idle resources to an endeavor for the convenient discovery of previously unknown truth.