## Friday, July 23, 2010

### 677

677 is a prime number.

677 has a representation as a sum of two squares: 677 = 12 + 262.

677 is the sum of three consecutive squares: 677 = 142 + 152 + 162.

677 is the hypotenuse of a primitive Pythagorean triple: 6772 = 522 + 6752. 677 Aaltje is an asteroid named for a Dutch singer.

#### 1 comment: Anonymous said...

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