167 (one hundred
ndsixty-seven) is the
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
following
166 and preceding
168.
In mathematics
167 is the 39th prime number, an
emirp, an isolated prime, a
Chen prime, a
Gaussian prime
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
, a
safe prime, and an
Eisenstein prime with no imaginary part and a real part of the form
.
167 is the smallest number which requires six terms when expressed using the greedy algorithm as a sum of squares, 167 = 144 + 16 + 4 + 1 + 1 + 1,
although by
Lagrange's four-square theorem
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number, nonnegative integer can be represented as a sum of four non-negative integer square number, squares. That is, the squares form an additive basi ...
its non-greedy expression as a sum of squares can be shorter, e.g. 167 = 121 + 36 + 9 + 1.
167 is a
full reptend prime
In number theory, a full reptend prime, full repetend prime, proper primeDickson, Leonard E., 1952, ''History of the Theory of Numbers, Volume 1'', Chelsea Public. Co. or long prime in base ''b'' is an odd prime number ''p'' such that the Fermat ...
in base 10, since the decimal expansion of 1/167 repeats the following 166 digits: 0.00598802395209580838323353293413173652694610778443113772455089820359281437125748502994 0119760479041916167664670658682634730538922155688622754491017964071856287425149700...
167 is a
highly cototient number, as it is the smallest number ''k'' with exactly 15 solutions to the equation ''x'' - φ(''x'') = ''k''. It is also a strictly non-palindromic number.
167 is the smallest multi-digit prime such that the product of digits is equal to the number of digits times the sum of the digits, i. e., 1×6×7 = 3×(1+6+7)
167 is the smallest positive integer ''d'' such that the imaginary
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of Degree of a field extension, degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free ...
Q() has
class number = 11.
External links
Prime curiosities: 167
References
{{DEFAULTSORT:167 (Number)
Integers