The Erdős distinct distances problem asks for the smallest number of distinct positive Euclidean
distances determined by distinct points in the plane
(Erdős 1946). The conjectured order of growth is
Guth and Katz (2015) proved the lower bound for an absolute constant
and .
The remaining logarithmic gap is open (Bloom 2026). For , 2, ..., the exact values begin 0, 1, 1, 2, 2, 3, 3, 4,
4, 5, 5, 5, 6, ... (OEIS A186704).
of distinct positive Euclidean
distances determined by 
distinct points in the plane
(Erdős 1946). The conjectured order of growth is
denotes asymptotic notation and 
is the natural logarithm.
A square lattice gives the upper bound of this order.
for an absolute constant 
and 
.
The remaining logarithmic gap is open (Bloom 2026). For 
, 2, ..., the exact values begin 0, 1, 1, 2, 2, 3, 3, 4,
4, 5, 5, 5, 6, ... (OEIS A186704).
Points." Amer. Math. Monthly 53, 248-250,
1946.