1051
This number is a prime.
C. J. Mozzochi utilized 1051 to show that if we let p(n)
be the nth prime, then there exists a constant K such that
p(n+1) - p(n) is less than K p(n)^(1051/1920). [Post]
1051 = π(1*1051) + π(0*1051) + π(5*1051) + π(1*1051). [Aghasoltan]
The largest known number n such that there is no prime between nk and (n+1)k, where k = 3/2. [Capelle]
The sum of all odd semiprimes less than a hundred. [Silva]
A book known as "The Great Omar" sank with the Titanic and has not been recovered. It contained 1051 precious and semi-precious stones set in eighteen-carat gold.
Before the Great Recession of the 1970's, the Dow Jones
Industrial Average peaked at 1051 on January 11, 1973.
(This is a record that would stand for nearly ten years.) [Cuenta]
There are at least 1051 cases in which the nth Honaker prime has the same number of digits as the nth square. The smallest are 4397 and 1024 (32^2). The largest known are 1066157 and 9998244 (3162^2). Could there possibly be more? [Gaydos]