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May 4th (125th Day of the Leap Year)

May 4th, 5/4.

125 = 5^(2+1)

125 = 152 - (5-2)^(2+1)

125 = 215 - 21*5 + 15

125 = 1*25 + 12*5 + (1+2+5)*(5*(2-1))

125 = 5*21 + 52*1 - 1*2^5

125 = (1+2+5)*((1+2)*5) - (1-2)*5

125 = (1*2*5)*(2*(1+5)) + 5*(2-1)

125 = (1+25)*(5*(2-1)) - ((5+1)/2 + (5-2-1))

125 = (12+5)*(2+5) + (5+2-1)

125 = (52+1)*(5-2-1) + (21-(5-2-1))

125 = 2^(5+1) + (52+(5*2-1))

125^n ends in "25" for all positive integers n. (This works for any number ending in 25).

125*521 contains 125. (What's the next number with the same property?)

125 = 5^3.

1+2+5 = 2^3.

(Are there other numbers with both these properties?)

There are 125 primes less than 700.

125^2 + 4 is prime (Note: 125 is of the form p^2 + 4 for prime p as well). (What's another number with both these properties?)

125 equals the third prime to the third power.

There are 125 primes between 19^3 and 20^3.

There are 39 primes between 125^2 and 126^2 and 39 primes between 125 and 2*125.