回复：有没有好办法？

The third side c is odd, let c=2x-1,

c>3, c<133, so 2<x<67

s=(2x-1+68+65)/2=x+66

 

S²=(x-2)*(x+1)*(67-x)*(66+x)

 

Possible Commun Divisor	x-2	x+1	67-x	66+x
x-2		3	5, 13	2, 17
x+1			2, 17	5, 13
67-x				7,19
66+x

 

1) x-2=3n

s=66+x = 68+3n , s cannot be divided by 3.

s cannot divided neither by 2 nor by 13*. all possible factors of s are 19, 17, 7, 5

a.       s=19m (3<m<7), so m=5

m=5, x=29, x-2=27, x+1=30, 67-x=38, 66+x=95

c=57, S²=(3*9)*(3*2*5)*(2*19)*(19*5), S=3*3*2*5*19

b.      s=17m  (4<m<8),  m=5, 7

m=5, x=9, x+1=10=5*2

m=7, x=53, x-2=51=3*17, x+1=54, 67-x=14, 66+x=119

c=105, S²=(3*17)*(3*2*9)*(2*7)*(17*7), S=3*2*3*7*17

c.       s=7m (9<m<19), m=17, idem b

d.      s=5m (13<m<27), m=19 idem a.

 

2) x-2 =3n+1,

66+x = 3* (23+n) è 66+x= 9m (7<m<14), m=8,9,10,13

m=8, x=6, 67-x=61

m = 9, x=15, x-2=13, x+1=16, 67-x=52, 66+x=81

c=29,  S²=(13)*(4*4)*(2*2*13)*(9*9), S=13*4*2*9

m=10, x=24, 67-x=43

m=13, x=51, x-2=49, x+1=52, 67-x=16, 66+x=117

c=101, S²=(7*7)*(2*2*13)*(4*4)*(13*9), S=7*2*13*4*3

 

3) x-2 =3n+2

67-x = 3*(21-n) à 67-x = 9m (1<m<8), so = 1, 2, 4, 5, 7

m=1, x=58, x+1=59

m=2, x=49, x-2=47

m=4, x=31, x-2=29

m=5, x=22, x+1=23

m=7, x=4, x-2=2, x+1=5, 67-x=63, 66+x=70

c=7, S²=(2)*(5)*(7*9)*(7*2*5), S=2*5*7*3 = 210

 

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x cannot be divided by 2 neither

if x is even, then n is also even , n=2m,

         then x-2=3*2m, x+1=3(2m+1)

         m=3p

                  x-2= 2*9p, x+1= 2*9p+3,

                  p must be divided by 3, so p= 3, x=56,

                  67-x = 11

          m=3p+1,

                  x+1=6m+3=18p+9=9(2p+1), x-2=6(3p+1)

                  2p + 1 must be divided by 3, so p=1, x=26,

                  x+66=92=4*23

           m=3p +2 

                  x-2=9p+6, x+1= 9p+9 = 9(p+1)

                  p+1 must be divided by 3, so p=2, x=26,

                  67-x=41

 

s cannot be divided by 13, otherwise x+1 can also be divided by 13 (see table)

x+1=13m,

x-2=3n =13m-3, m can not be odd, otherwise x-2 and x are both even number.

The smallest m is 6 which gives x=77, too big.