167 (number)

167 is the 39th prime number, an emirp, an isolated prime, a Chen prime,{{Cite OEIS|A109611|Chen primes: primes p such that p + 2 is either a prime or a semiprime}} a Gaussian prime, a safe prime,{{Cite OEIS|A005385|Safe primes}} and an Eisenstein prime with no imaginary part and a real part of the form $3n\; -\; 1$.

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,{{Cite OEIS|A006892|name=Representation as a sum of squares requires n squares with greedy algorithm}}

although by Lagrange's four-square theorem 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 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 Q({{sqrt|–d}}) has class number = 11.{{cite web|title=Tables of imaginary quadratic fields with small class number|website=numbertheory.org|url=http://www.numbertheory.org/classnos/}}

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• [http://primes.utm.edu/curios/page.php/167.html Prime curiosities: 167]

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Category:Integers