Exotic Mathematical Constants
These important mathematical constants
are much less pervasive than the above ones...
The Delian constant is the scaling factor which doubles a volume.
The cube root of 2 is much less commonly encountered than
its square root (1.414...).
There's little need to remember that it's roughly equal to 1.26
but it can be useful
(e.g., a 5/8" steel ball weight almost twice as much as a 1/2" one).
The fact that this quantity cannot be constructed "classically"
(i.e., with ruler and compass alone)
shows that there's no "classical" solution to the so-called
whereby the Athenians were asked by the
Oracle of Apollo at Delos to resize the altar of Apollo
to make it "twice as large".
The Delian constant has also grown to be a favorite
example of an algebraic number of degree 3
(arguably, it's the simplest such number).
Thus, its continued fraction expansion
(CFE) has been under considerable scrutiny...
There does not seem to be anything special about it, but the question remains
(by contrast, the CFE of any algebraic number of degree 2 is
Gauss's constant (G) is the reciprocal of
On May 30, 1799, Carl Friedrich Gauss
found the following expression to be equal to the reciprocal of the
arithmetic-geometric mean between
1 and Ö2.
|G (¼) 2
The continued fraction expansion (CFE) of G is:
G = [ 0; 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, 3, 8, 36, 1, 2, 5, 2, 1, 1 ... ]
The symbol G is also used for Catalan's constant
which is best denoted b(2)
whenever there is any risk of confusion.
The limit of [1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + ... + 1/p] - ln(ln p)
Mertens constant has been named after the number theorist
It is to the sequence of primes what
Euler's constant is to the sequence of integers.
It's sometimes also called Kronecker's constant
or the Reciprocal Prime Constant.
Proposals have been made to name this constant
after Charles de la Vallée-Poussin (1866-1962) and/or Jacques Hadamard (1865-1963),
the two mathematicians who first proved (independently)
the Prime Number Theorem, in 1896.
(2006-06-15) Artin's Constant :
C = 0.373955813619202288054728+
The product of all the factors [ 1 - 1 / (q2- q) ]
for prime values of q.
For any prime p besides 2 and 5, the decimal
expansion of 1/p has a period at most equal
to p-1 (since only this many different nonzero "remainders" can possibly
show up in the long division process).
Primes yielding this maximal period are called
long primes [to base ten] by recreational mathematicians and others.
The number 10 is a primitive root modulo such a prime p,
which is to say that the first p-1 powers of 10 are
distinct modulo p (the cycle then
repeats, by Fermat's little theorem).
Putting a = 10, this is equivalent to the condition:
a (p-1)/d ¹ 1
(modulo p) for any prime factor d of (p-1).
For a given prime p, there are f(p-1)
satisfactory values of a (modulo p),
is Euler's totient function.
Conversely, for a given integer a,
we may investigate the set of
long primes to base a...
It seems that the proportion C(a) of such primes
(among all prime numbers) is equal to the above numerical
constant C, for many values of a
(including negative ones) and that it's always a
rational multiple of C.
The precise conjecture tabulated below originated with
Artin (1898-1962) who communicated it to
in September 1927.
Neither -1 nor a
quadratic residue can be a
primitive root modulo p > 3.
Hence, the table's first row is as stated.
conjecture for primitive roots (1927)
first refined by
(For a given "base",
just apply the earliest applicable case, in the order listed.)
of primes p for which
a is a primitive root
|-1 or b 2||0
|a = b k||C(a) =
v is multiplicative:
v(qn ) = q(q-2) / (q2-q-1) if q is prime
mod 4 = 1
See notation below*
[ 1 -
|1 + q - q2
|q | sf (a)
C(a) = C =
This last case applies to all integers, positive
or negative (A120629)
that are not perfect powers and whose
squarefree part isn't congruent to 1 modulo 4, namely :
2, 3, 6, 7, 10, 11, 12, 14, 15, 18, 19, 22, 23, 24, 26, 28,
30, 31, 34, 35, 38, 39, 40 ...
-2, -4, -5, -6, -9, -10, -13, -14, -16, -17, -18, -20, -21, -22, -24, -25,
-26, -29, -30, -33
(*) In the above, sf (a) is the
squarefree part of a,
namely the integer of least magnitude which makes the product
a sf (a) a square.
The squarefree part of a negative integer is the opposite of the
squarefree part of its absolute value.
The conjecture can be deduced from its special case about
prime values of a,
which states the density is C unless a
is 1 modulo 4, in which case it's equal to:
[ ( a 2 - a ) /
( a 2 - a - 1 ) ] C
In 1984, Rajiv Gupta and M. Ram Murty showed Artin's conjecture to be true
for infinitely many values of a.
In 1986, David Rodney ("Roger")
that there are at most 2 primes for which it fails...
Yet, we don't know about any
single value of a for which the result is certain!
Ramanujan-Soldner constant, root of the logarithmic integral:
li(m) = 0
This number is named after
Johann von Soldner
Srinivasa Ramanujan (1887-1920).
It's also called Soldner's constant.
m is the
only positive root of the logarithmic integral function "li",
which is not to be confused with the so-called offset logarithmic integral:
Li(x) = li(x)-li(2).
The above integrals must be understood as
Cauchy principal values
whenever the singularity at t = 1 is in the interval of integration...
The Omega constant.
This is the solution of x = e-x,
or also x = ln(1/x).
See Lambert's W function.
The value of the constant could be obtained by iterating
the function e-x, but the convergence is quite slow.
It's much better to iterate the function
f (x) = (1+x) / (1+ex )
which has the same fixed point but features a zero
derivative at this fixed point,
so that the convergence is quadratic
(the number of correct digits is roughly doubled
with each iteration).
This is an example of
What's known as the [first] Feigenbaum constant
is the "bifurcation velocity" (d)
which governs the geometric onset of chaos via period-doubling
in iterative sequences (with respect to some parameter
which is used linearly in each iteration, to damp a given function
having a quadratic maximum).
This universal constant was unearthed in October 1975 by
J. Feigenbaum (b.1944).
The related "reduction parameter" (a) is the
second Feigenbaum constant...
Feigenbaum Constant :
(Eric W. Weisstein)
Mathsoft (Steve Finch)