This number is a prime.
If we are to append to prime 1061 another smallest prime so that the concatenation would yield a square then this second prime would be 10601. Note that 10601 is 1061 with just an extra zero inserted. Also the square root of 106110601 is 10301. Both 10301 and 10601 are palindromic primes! [De Geest]
A four-digit prime that equals the number of primes with four digits. [Russo]
The first 1061 digits in the decimal expansion of the square root of 10 is prime. [Beedassy]
A "Bemirpyramid" of length 6: 1061 108810610081 11811088106100819881 111861181108810610081988181691 100991111861181108810610081988181691616681 100196100991111861181108810610081988181691616681996061
Found by Metin Sariyar of Istanbul, Turkey.
A monk spotted Halley's Comet in 1061 and predicted the destruction of England. [Couper]