These numbers are defined similarly to the fibonomials, but involve terms of the form F_Fn, where F_n represents the n^th Fibonacci number. For example, the 4^th element in the 7^th row (210) is F_F6F_F5F_F4/F_F3F_F2F_F1 = F₈F₅F₃/F₂F₁F₁ = 21×5×3/1×1×1. The general form is *F_Fn...FFn-k+1/F_Fk*...FFv{1. This formula always gives an integer, for reasons explained in the fibonomial description.

The second number on each row is the sequence of F_Fn (Sloane's A007570): 1,1,1,2,5,21,233,10946,5702887,.... The third number on each row is F_Fn×F_Fn-1, a sequence that starts: 1, 2, 10, 105, 4893, 2550418, 62423801102, ... The 4^th number in each row is F_Fn×F_Fn+1×F_Fn+2/2, a sequence that starts 1, 2, 10, 210, 24465, 53558778, ....

496 = 2⁴ (2⁵-1)

The third perfect number. The even perfect numbers (it is not known if there are any odd perfect numbers) can all be expressed in the forms:

2^P-1 (2^P - 1)

2^N (2^N+1 - 1) where P is a prime and N = P+1. In this example, P is 17. Also, for the number to be perfect, 2^P-1 must be prime, and is called a Mersenne prime. See here for a complete list of known perfect numbers.

See also 6, 28, 8589869056, and 4.4823309×10^14471464.

561 = 3×11×17

The first Carmichael number. Carmichael numbers are odd composite numbers which cannot be found to be composite using Fermat's Little Theorem, which states that for all prime numbers p, if n is relatively prime to p, then n^p-1-1 is divisible by p (sometimes stated: n^p - n is divisible by p, which is an equivalent statement). For a Carmichael number, no matter what n you pick this test will show that it might be prime. See also 1729.

611

Common letter-number value of the word Torah in Hebrew. There are several Hebrew alphabetic number assignments used for Gematria (numerology) but only one system for the common purpose of communicating a number in ordinary text. It dates back to before they used separate symbols for the digits the way they do now. The assignment is as follows¹⁵:

א=1 ב=2 ג=3 ד=4 ה=5 ו=6 ז=7 ח=8 ט=9 י=10 כ=20 ל=30 מ=40 נ=50 ס=60 ע=70 פ=80 צ=90 ק=100 ר=200 ש=300 ת=400

Using that assignment, the value of תורה (Torah) is 611 (400+6+200+5)

616

Appears instead of 666 in some early Greek manuscripts of the relevant passage of the Bible. See the 666 entry for more.

641

The first factor of a non-prime Fermat number. Fermat conjectured that the Fermat numbers were all prime, and could not factor 2³²+1=4294967297. Euler showed that all factors of a Fermat number 2²ⁿ+1 must be of the form K2ⁿ⁺¹+1. In this case, n is 5 and the factor 641 is equal to 10×2⁵⁺¹+1. For more about this see the Wiki page on Fermat primes.

648

648 is the smallest number that can be expressed as a b^a in two different ways: 3×6³ = 2×18². Such numbers are not particularly common: 648, 2048, 4608, 5184, 41472, 52488, 472392, 500000, 524288, 2654208, 3125000, 4718592, ...

648 = 3×6³ = 2×18²
2048 = 8×2⁸ = 2×32²
4608 = 9×2⁹ = 2×48²
5184 = 4×6⁴ = 3×12³
41472 = 3×24³ = 2×144²
52488 = 8×3⁸ = 2×162²
472392 = 3×54³ = 2×486²
500000 = 5×10⁵ = 2×500²
524288 = 8×4⁸ = 2×512²
2654208 = 3×96³ = 2×1152²
3125000 = 8×5⁸ = 2×1250²
4718592 = 18×2¹⁸ = 2×1536²
10125000 = 3×150³ = 2×2250²
13436928 = 8×6⁸ = 2×2592²
21233664 = 4×48⁴ = 3×192³
30233088 = 3×216³ = 2×3888²
46118408 = 8×7⁸ = 2×4802²
76236552 = 3×294³ = 2×6174²
134217728 = 8×8⁸ = 2×8192²
169869312 = 3×384³ = 2×9216²
344373768 = 8×9⁸ = 3×486³ = 2×13122²
402653184 = 24×2²⁴ = 3×512³
512000000 = 5×40⁵ = 2×16000²
648000000 = 3×600³ = 2×18000²
737894528 = 7×14⁷ = 2×19208²
800000000 = 8×10⁸ = 2×20000²
838860800 = 25×2²⁵ = 2×20480²
922640625 = 5×45⁵ = 3×675³
1147971528 = 3×726³ = 2×23958²
1207959552 = 9×8⁹ = 2×24576²
1714871048 = 8×11⁸ = 2×29282²
1934917632 = 3×864³ = 2×31104²
2754990144 = 4×162⁴ = 3×972³
3127772232 = 3×1014³ = 2×39546²
3439853568 = 8×12⁸ = 2×41472²
4879139328 = 3×1176³ = 2×49392²
6525845768 = 8×13⁸ = 2×57122²
6973568802 = 18×3¹⁸ = 2×59049²
7381125000 = 3×1350³ = 2×60750²

See also 344373768.

666

The mother of all cult numbers, as described below — but first I will list the genuinely mathematical properties of 666:

666 is the sum of the squares of the first 7 primes: 2² + 3² + 5² + 7² + 11² + 13² + 17².

Another similar sum: 666 = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 5³ + 4³ + 3³ + 2³ + 1³. Such sums could be called "hyper-octahedral" numbers, based on a 4-dimensional polyhedron analagous to the octahedron. Perhaps when Christian numerologists go to heaven, instead of a one-mile cube, their new Jerusalem will be a 6-mile hyperoctahedron.

It is the 36th triangular number: 1+2+3+...+36 = 666, which seems more significant because 36=6×6. 666 is the largest triangular number that is a "repdigit" (all digits the same).

666 = 3⁶-2⁶+1⁶.

Φ(666), the number of integers less than and relatively prime to 666, is 6×6×6 = 216.

If you take the prime factors of 666 = 2×3×3×37 and add their digits, you get 18 = 6+6+6. That makes 666 the 34^th Smith number. That's not too special — it's true for 49 of the first 1000 composite numbers.

As history's greatest cult number, 666 has attracted more attention than nearly any other number in numerology, for at least several hundred years if not longer. It is the "number of the beast" mentioned in the book of Revelation (the only apocalypse in the Christian canon, attributed to the apostle John). The relevant verse is Rev 13:18; in the original Greek it reads:.

ψδε η ςουια εςσιν. ο εφψν νοτν Πηυιςασψ σον αρι+μον σοτ +ηριοτ͵ αρι+μοΣ γαρ αν+ρψποτ εςσιν͵ και ο αρι+μοΣ ατσοτ εξακοςιοι εξηκονσα εξ.
Phonetically: Ode E sophia estin o echOn. noun psEphisatO ton arithmon tou thEriou, arithmos gar anthrOpou estin, kai o arithmos autou exakosioi exEkonta ex.

Translations vary because they attempt to make passages sound natural and avoid cumbersome constructions. However, the most accurate translations resemble this (from the 1958 Amplified New Testament):

Here is [room for] discernment [a call for the wisdom of interpretation]. Let anyone who has intelligence (penetration and insight enough) calculate the number of the beast, for it is a human number [the number of a certain man]; his number is six-hundred sixty six.

εξακοςιοι εξηκονσα εξ is "six-hundred sixty six". Notably, some early Greek texts²⁰ read "six-hundred ten six", that is, 616. Why was the specific number 666 (or 616) chosen for this passage? The wording clearly indicates that some effort and calculation will be needed, and that something isn't directly obvious without applying this "wisdom". Since it says "calculate this number" in one place, but then goes on to say "[the] number is 666", there is a contradiction: what would the reader have to calculate if the writer gives away the answer in the next phrase? There are two ways to interpret the contradiction: either the number itself is unspecified (and 666 is just a metaphor), or else 666 is indeed the answer but that the method of calculation to produce it (and/or perhaps the identity of the "certain man") is unspecfied.

David Wells⁴ suggests the first interpretation, when he points out the fact that 666 in Roman numerals (DCLXVI) was often used as a generic way of referring to any unspecified or unknown large number. (616 would have been similar, DCXVI). "DCLXVI" is a Roman slang way of saying "any big long number", like our colloquial slang zillion, because big numbers in Roman numerals always end up being long unmanagable strings with lots of different letters in them. Thus, the writer of Revelation might have been using "666" to mean "an unspecified or unknown large number". This makes sense because of the fact that the writer says wisdom is needed to calculate the number: if he were giving the number explicitly, you wouldn't need any wisdom to calculate it. He is saying that the number of the beast is large, unknown and unspecified, but if you have wisdom or insight you might be able to figure out what it is.

Supporting this general idea, numerologists like to point out that any of our common numbers (telephone, license plate, credit cards, tax ID numbers, etc.) would have appeared very large to someone living 2000 years ago. Indeed, they are too large to represent in most of the ancient number systems, so it would have been impossible to know or to specify such numbers. A very great wisdom indeed would have been required to calculate one!

The other possible interpretation, namely that the answer actually is 666 but that you have to discover what calculation will yield that answer (or which "certain man" owns the number), is the one that has drawn the most attention. An incredible variety of attempted calculations, connections, interpretations, hidden meanings, etc. have been found for the number 666, and a fair number for 616 as well. They have plenty to go on, given the availability of no less than four relevant contemporary languages (Hebrew, Aramaic, Greek, Latin) each with a variety of different numerical and numerological systems. There is clear evidence (in the grammar and vocabulary) that Revelation was translated from Hebrew to Greek, or written by someone for whom Greek was a second language.

numerology

Numerology connects a number or multiple numbers with something else, such as a person's name or an important word, phrase or passage of text. The determination of which number(s) to use generally follows some type of letter-value system. For example, the Greek system is shown here; each letter is turned into a number and then added together.

In Hebrew culture this type of numerology is part of Gematria. The numbers reveal otherwise-hidden meanings and connections between one name, phrase, etc. and another. The Hebrew alphabet has 22 letters, and there are many different systems for assigning numbers to the letters for Gematria purposes. There is also one numbering system (shown here) used for everyday purposes. Using that system, here is an example:

Saddam = סאדאם (samex, alef, dalet, alef, mem(final)) = 60 + 1 + 4 + 1 + 600 = 666
Hussein = הוששהנ (he, vau, shin, shin, he, nun) = 5 + 6 + 300 + 300 + 5 + 50 = 666

The name "Saddam Hussein" has been spelled as it would be in Hebrew and the values of the letters are added to get the sums. The second line is sort of cheating, because the נ (nun), being the final letter, should be a final nun ן, and its value would then be 700, not 50. (Or alternately, change the mem at the end of Saddam to a non-final mem מ with value 40). Such "cheating" is widespread in numerology particularly when a connection to 666 is being shown, and the numerologist always finds a reason to justify bending the rules.

An equivalent letter-numbering system, adapted to English, would go something like this:

A=1 B=2 C=3 D=4 E=5 F=6 G=7 H=8 I=9 J=10 K=20 L=30 M=40 N=50 O=60 P=70 Q=80 R=90 S=100 T=200 U=300 V=400 W=500 X=600 Y=700 Z=800

Some examples:

FOX = 6 + 60 + 600 = 666
GUTENBERG = 7 + 300 + 200 + 5 + 50 + 2 + 5 + 90 + 7 = 666
NEWARK = 50 + 5 + 500 + 1 + 90 + 20 = 666

Naturally, our high-tech age has given numerologists plenty of new ways to compute these sums. Computers give each letter, digit and symbol a number according to the ASCII code. For example, "MS-DOS 6.21" = 77 + 83 + 45 + 68 + 79 + 83 + 32 + 54 + 46 + 50 + 49 = 666.

Simpler techniques have been used as well, such as counting the number of letters in each word: "Ronald Wilson Reagan" (6 letters each) = 6 6 6.

Many numerologists like to prove things based on more pure-mathematical properties (that is, not depending on words or alphabets). For example, if you take 666⁷ = 58,119,196,856,185,328,256 and add the digits in groups of 3, the total is 1998. The same is true for a few other powers of 666. (This was used as part of an elaborate and somewhat illucid discussion on this web page, which for a while was titled Proof That the World Will End In 1998. It features lots of really amusing detail, such as poetry allegedly translated from Aramaic but which rhymes in English, and of course an update appropriately titled Why the World Did Not End In 1998 (the short answer: the world is ending, you just didn't notice!).) As it happens, all powers of 666 (starting with 666²) are multiples of 999, and therefore, because of the casting out 9's property applied to 999, all sums of 3-digit groups of powers of 666 are multiples of 999. Thus, many powers of 666 all have the same 3-digit-group sums; the same thing happens for powers of 333 and 999 for the same reasons.

The claim that 39 is uninteresting goaded one numerologist into showing that 39 is related to 666 in the following way:

39² = 1521
15² + 21² = 666

See also 10^{(6.65565×10668)}

676

This is 26², the first square that is a palindrome but whose square root is not a palindrome.

In the English language there are 26 letters, so 676 is the number of combinations of 2 letters when order is distinct. If the standard license plates in your area had 2 letters followed by 4 numbers, there would be 26²×10⁴ = 676×10000 = 6760000 possible plates.

695

According to my classical sequence generator, 695 is the next number after my childhood "favorite" numbers 7, 27 and 127. The formula it finds is: A₀ = -1; A₁ = 1; A_N+1 = N A_N + 2A_N-1 + N + 1 (sequence MCS30054923). This serves as an example of how easy it is to find a sequence formula to match an arbitrary set of numbers. See also 1011.

714

The first 7 prime numbers (2, 3, 5, 7, 11, 13, 17) can be arranged to form the factors of 714 (2×3×7×17) and 715 (5×11×13). Because of this, in base 714 there are easy tests for divisibility by 9 different primes (the 7 just listed plus 23 and 31 because 23×31=713). See the 14, 21, 29 and 66 entries for more about these properties.

714 and 715 also have the property that their prime factors add up to the same total: 2 + 3 + 7 + 17 = 5 + 11 + 13. Another (smaller) example is 77 and 78: 77=7×11, 78=2×3×13, and 7+11=2+3+13. When one of the pair is divisible by a square, it matters how you count the factors. For example, 24 and 25 are a pair if you count each prime only once: 24=2×2×2×3, 25=5×5, and 2+3=5. 15 and 16 qualify if you count the multiples: 15=3×5, 16=2×2×2×2, and 3+5=2+2+2+2. But the pair 714 and 715 qualify regardless of which way you define it. Such pairs are called Ruth-Aaron pairs because of Babe Ruth's famous record of 714 career home runs, which was broken in 1974 when Hank Aaron hit his 715^th. (Aaron reached 755 before retiring, but the number 715 is almost as commonly associated with him; in 2007 his record was surpassed by Bonds).

720 = 6!

Like 120 and 210, 720 can be expressed as a product of consecutive integers in two distinct ways: 2×3×4×5×6 = 8×9×10. You can also add a 7 to both sides and get a similar equation whose value is 5040. Here are the smallest numbers with this property: 120, 210, 720, 5040, 175560, 17297280, 19958400, 259459200, 20274183401472000, 25852016738884976640000, 368406749739154248105984000000, ... The sequence is Sloane's A100933. The sequence is infinite — see 19958400 for details as to why. I also have a page dedicated to these numbers.

757

757 is the smallest factor of 999999999999999999999999999=10²⁷-1 that is not also a factor of a smaller string of 9's, and therefore 757 is the smallest number whose reciprocal has a 27-digit repeating decimal: 1/757=.00132100396301188903566710700132... This is part of a series: 1/3 has a 1-digit pattern, 1/27 has a 3-digit pattern, and 1/757 has a 27-digit pattern. The series keeps going up (because if 1/n has a d-digit pattern, n is always larger than d), but computing the next number in the series is hard because it is equivalent to factoring a large number like 10⁷⁵⁷-1. See also 7, 27 and 239.

768

Like 6, 12, 24, 48, 96, 192 and 384, 768 is a 3-smooth number of the form 3×2ⁿ. It is one of several such numbers to occur in personal computer display dimensions (the standard 1024x768 "XGA" mode); see also 192.

784

See 82944.

840

840 is a factors record-setter, and is the first that does not also appear as the number of factors of another factors record-setter. See also 12, 45360, 720720, 3603600, 245044800, 278914005382139703576000, 2054221614063184107682218077003539824552559296000 and 457936×10⁹¹⁷.

841

841 = 29² is the sum of two consecutive squares: 20²+21². This is the second smallest example; the smallest is 5² = 25 = 3²+4². The series continues: 5², 29², 169², 985², 5741², 33461², 195025², ... (Sloane's integer sequence A001653). This sequence consists of every other Pell number; Each of these is 6 times the previous one minus the one before that, for example, 169 = 6×29-5. See also 99 and 204.

952

952 = 9³+5³+2³+9×5×2. (Thanks to Cyril Soler for this tip)

998

1/998 = 0.001002004008016032064128256513026052104208416833667334669..., a repeating decimal in which the powers of 2 appear one after another (until they start to overlap and break the pattern)

0.001
0.000002
0.000000004
0.000000000008
0.000000000000016
0.000000000000000032
0.000000000000000000064
0.000000000000000000000128
0.000000000000000000000000256
0.000000000000000000000000000512
0.000000000000000000000000000001024
0.000000000000000000000000000000002048
. . .
0.001002004008016032064128256513026052...

The same thing happens to a lesser degree in the digits of 1/98, and to a greater degee in the digits of 1/9998, 1/99998, etc. A similar pattern involving the Fibonacci numbers appears in the reciprocal of 89.

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