289

289 = 17².

289 is the largest 3-digit square with increasing digits.

289 has a representation as a sum of two squares: 289 = 8² + 15².

289 is the hypotenuse of a primitive Pythagorean triple: 289² = 161² + 240².

289 is a centered octagonal number.

The Ford Mustang 289 was a high-performance version of the car in the 1960s.

Source: Number Gossip

625

625 = 5⁴. It is a perfect fourth power.

Two fourth powers use the same digits: 5⁴ = 625 and 4⁴ = 256.

625 has two representations as a sum of two squares: 625 = 7² + 24² = 15² + 20².

625 is the hypotenuse of a primitive Pythagorean triple: 625² = 336² + 527².

625 is a centered octagonal number.

625 is an automorphic number.

According to a 2010 White House news release, 625 people lost their health insurance every hour in 2009.

Source: What's Special About This Number?