**677**is a prime number.

**677**has a representation as a sum of two squares:

**677**= 1

^{2}+ 26

^{2}.

**677**is the sum of three consecutive squares:

**677**= 14

^{2}+ 15

^{2}+ 16

^{2}.

**677**is the hypotenuse of a primitive Pythagorean triple:

**677**

^{2}= 52

^{2}+ 675

^{2}.

**677**Aaltje is an asteroid named for a Dutch singer.

Source: Prime Curios!

## 1 comment:

Other properties of the number 677

677 can be represented as 677=3^6-52

677 divides 26^4-1

677 can be represented as other Pythagorean triples: 677^2=70200^2 +452921^2 or 677^2+229124^2=229125 ^2

677 Fermat's cousin and that 677=2k+1=2*338 and 677=4k+1=4*169. It is a number that, as mentioned, can be represented as the sum of two squares, then, is a Gaussian integer, then

677=(26+i)(26-i) as a reciprocal or conjugate or

677=(26+i)(1+26i)(-i) as in factorization with symmetric unit

Using the conjecture of Albert Girard that every number can be a square sum of plus or minus another number multiplied by a square of 2 or more a square-minus the sum of prime, we can establish that, 677 has the representation:

677=4^2+661=6^2+677=8^2+613=10^2+577

677=3^2+167*2^2=5^2+163*2^2=7^2+153*2^2=9^2+149*2^2

or

677=28^2-107=30^2-223=32^2-347=34^2-479 =36^2-619

677=27^2-13*2^2=29^2-41*2^2 =31^2-71*2^2=33^2-103*2^2=35^2-137*2^2

677 is a typical number that allows a teacher to develop a masterclass on the unique factorization in quadratic fields.

A curiosity: The Angelo's 677 Prime in Albany a restaurant, view

http://www.677prime_wedding_events

Rafael Parra DE Barcelona

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