Pseudo Pythagorean Triples
Consider the subset of Pythagorean triples, a^2+b^2=c^2, where:
1) for every a, where a is an odd number (greater than 1)
2) there is a b,c such that b and c are consecutive integers (c-b=1) and b+c=a^2
For example:
a=3, b=4, c=5 ; 5-4=1 ; 4+5 = 9 = 3^2 and 3^2+4^2=5^2
a=5, b=12, c=13 ; 13-12=1 ; 12+13 = 25 = 5^2 and 5^2+12^2 =13^2
a=7, b=24, c=25 ; 25-24=1 ; 24+25 = 49 = 7^2 and 7^2+24^2=25^2
etc.
Now consider the following set of pseudo Pythagorean triples, 3(a^2)+b^2=c^2, where
1) for every a, where a is an odd number (greater than 1)
2) there is a b,c such that c-b=3 and (b+c)=a^2
For example:
a=3, b=3, c=6 ; 6-3=3 ; 3+6=9=3^2 ; 3(3^2)+3^2=36=6^2
a=5, b=11, c=14 ; 14-11=3 ; 11+14=25=5^2 ; 3(5^2)+11^2=196=14^2
a=7, b=23, c=26 ; 26-23=3 ; 23+26=49=7^2 ; 3(7^2)+23^2=676=26^2
This process can be adjusted to make other sets of pseudo Pythagorean triples, where a is an odd number (appropriately adjusted to be greater than 0 with respect to n), c-b=n (where n continues the series 1,3,5,7,9,etc.), and b+c=a^2, such that n(a^2)+b^2=c^2
For example: 5(a^2)+b^2=c^2
a=3, b=2, c=7 ; 7-2=5 ; 2+7=9 ; 5(3^2)+2^2=49=7^2
a=5, b=10, c=15 ; 15-10=5 ; 10+15=25 ; 5(5^2)+10^2=225=15^2
a=7, b=22, c=27 ; 27-22=5 ; 22+27=49 ; 5(7^2)+22^2=729=27^2
etc.
Now consider the set of pseudo Pythagorean triples , where a is an even number (appropriately adjusted to be greater than 0 with respect to n), c-b=n (where n is in the series 2,4,6,8,10,etc.), and b+c=a^2, such that n(a^2)+b^2=c^2
For example:
2(a^2)+b^2=c^2
a=2, b=1, c=3 ; 3-1=2 ; 1+3=4 ; 2(2^2)+1^2=9=3^2
a=4, b=7, c=9 ; 9-7=2 ; 7+9=16 ; 2(4^2)+7^2=81=9^2
a=6, b=17, c=19 ; 19-17=2 ; 17+19=36 ; 2(6^2)+17^2=361=19^2
a=8, b=31, c=33 ; 33-31=2 ; 31+33=64 ; 2(8^2)+31^2=1089=33^2
etc.
4(a^2)+b^2=c^2
a=4, b=6, c=10 ; 10-6=4 ; 6+10=16 ; 4(4^2)+6^2=100=10^2
a=6, b=16, c=20 ; 20-16=4 ; 16+20=36 ; 4(6^2)+16^2=400=20^2
a=8, b=30, c=34 ; 34-30=4 ; 30+34=64 ; 4(8^2)+30^2=1156=34^2
etc.
6(a^2)+b^2=c^2
a=4, b=5, c=11 ; 11-5=6 ; 5+11=16 ; 6(4^2)+5^2=121=11^2
a=6, b=15, c=21 ; 21-15=6 ; 15+21=36 ; 6(6^2)+15^2=441=21^2
a=8, b=29, c=35 ; 35-29=6 ; 29+35=64 ; 6(8^2)+29^2=1225=35^2
etc.
etc.
