# Math Is Fun Forum

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## #51 2024-06-06 23:25:46

Jai Ganesh
Registered: 2005-06-28
Posts: 47,007

### Re: Some special numbers

62) Eddington Number

In astrophysics, the Eddington number,

, is the number of protons in the observable universe. Eddington originally calculated it as about
; current estimates make it approximately

The term is named for British astrophysicist Arthur Eddington, who in 1940 was the first to propose a value of

and to explain why this number might be important for physical cosmology and the foundations of physics.

History

Eddington argued that the value of the fine-structure constant, α, could be obtained by pure deduction. He related α to the Eddington number, which was his estimate of the number of protons in the universe. This led him in 1929 to conjecture that α was exactly 1/136. He devised a "proof" that

. Other physicists did not adopt this conjecture and did not accept his argument.

In the late 1930s, the best experimental value of the fine-structure constant, α, was approximately 1/137. Eddington then argued, from aesthetic and numerological considerations, that α should be exactly 1/137.

Current estimates of

point to a value of about
. These estimates assume that all matter can be taken to be hydrogen and require assumed values for the number and size of galaxies and stars in the universe.

During a course of lectures that he delivered in 1938 as Tarner Lecturer at Trinity College, Cambridge, Eddington averred that:

I believe there are 15747724136275002577605653961181555468044717914527116709366231425076185631031296 protons in the universe and the same number of electrons.

This large number was soon named the "Eddington number".

Shortly thereafter, improved measurements of α yielded values closer to 1/137, whereupon Eddington changed his "proof" to show that α had to be exactly 1/137.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #52 2024-06-26 23:11:53

Jai Ganesh
Registered: 2005-06-28
Posts: 47,007

### Re: Some special numbers

63) Number 4

2^^2 = 4,
2^^^2 = 4,
2^^^^2 = 4,
2^^^^^2 = 4,
2^^^^^^2 = 4......

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #53 2024-06-27 20:32:13

Jai Ganesh
Registered: 2005-06-28
Posts: 47,007

### Re: Some special numbers

64) Powers

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #54 2024-06-27 20:52:02

Jai Ganesh
Registered: 2005-06-28
Posts: 47,007

### Re: Some special numbers

65)

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #55 2024-07-07 17:41:48

Jai Ganesh
Registered: 2005-06-28
Posts: 47,007

### Re: Some special numbers

66) 17

.

.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #56 2024-07-07 18:06:25

Jai Ganesh
Registered: 2005-06-28
Posts: 47,007

### Re: Some special numbers

67) Weird number

In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.

Examples

The smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70. The number 12, for example, is abundant but not weird, because the proper divisors of 12 are 1, 2, 3, 4, and 6, which sum to 16; but 2 + 4 + 6 = 12.

The first few weird numbers are

70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, ...

Properties

Infinitely many weird numbers exist. For example, 70p is weird for all primes p ≥ 149. In fact, the set of weird numbers has positive asymptotic density.

It is not known if any odd weird numbers exist. If so, they must be greater than

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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## #57 2024-07-08 19:51:31

Jai Ganesh
Registered: 2005-06-28
Posts: 47,007

### Re: Some special numbers

68) Achilles number

An Achilles number is a number that is powerful but not a perfect power. A positive integer n is a powerful number if, for every prime factor p of n,

is also a divisor. In other words, every prime factor appears at least squared in the factorization. All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as
, where m and k are positive integers greater than 1.

Achilles numbers were named by Henry Bottomley after Achilles, a hero of the Trojan war, who was also powerful but imperfect. Strong Achilles numbers are Achilles numbers whose Euler totients are also Achilles numbers; the smallest are 500 and 864.

The Achilles numbers up to 5000 are:

72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000.

The smallest pair of consecutive Achilles numbers is:

Examples

As an example, 108 is a powerful number. Its prime factorization is

, and thus its prime factors are 2 and 3. Both
are divisors of 108. However, 108 cannot be represented as
, where m and k are positive integers greater than 1, so 108 is an Achilles number.

The integer 360 is not an Achilles number because it is not powerful. One of its prime factors is 5 but 360 is not divisible by

Finally, 784 is not an Achilles number. It is a powerful number, because not only are 2 and 7 its only prime factors, but also

are divisors of it. It is a perfect power:

So it is not an Achilles number.

The integer

is a strong Achilles number as its Euler totient of
is also an Achilles number.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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