Math Newsletter number 19; Wednesday, December 1, 2010.
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Reducible 4th Degree polynomials

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p**4+p**2 q**2+q**4 = (p**2 - p q + q**2)(p**2 + p q + q**2)

1**4+(1**2)(1**2)+1**4 =(1**2 -1*1 +1**2)(1**2 + 1*1 + 1**2)

3 = 1 * 3

2**4 +(2**2)(1**2)+ 1**4=(2**2-2*1+1**2)(2**2+2*1+1**2)

21 = 3 * 7

3**4 +(3**2)(1**2)+ 1**4=(3**2-3*1+1**2)(3**2+3*1+1**2)

91 = 7*13

3**4+(3**2)(2**2)+2**4=(3**2-3*2+2**2)(3**2+3*2+2**2)

133 = 7*19

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4 p**4 + q**4 = (2 p**2 -2 p q +q**2)(2 p**2 +2 p q + q**2)

4*(1**4)+1**4 = (2(1**2)-2*1*1+1**2)(2(1**2)+2*1*1+1**2)

5 = 1*5

4*(2**4)+1**4=(2(2**2)-2*2*1+1**2)(2(2**2)+2*2*1+1**2)

65 = 5*13

4*(3**4)+1**4=(2(3**2)-2*3*1+1**2)(2(3**2)+2*3*1+1**2)

325 = 13 * 25

4*(1**4)+3**4=(2(1**2)-2*1*3+3**2)(2(1**2)+2*1*3+3**2)

85 = 5 * 17

4* 1**4 + 5**4 = (2 * 1 - 2*1*5 + 5**2)(2*1 + 2*1*5 + 5**2)

629 = 17 * 37

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9 p**4+2p**2 q**2+q**4=(3p**2 -2pq+q**2)(3p**2+2pq +q**2)

9*1+2*1*1+1 = (3*1-2*1*1+1)(3*1+2*1*1+1)

12 = 2 * 6

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9p**4+3p**2 q**2 +4 q**4=(3p**2-3pq+2q**2)(3p**2+3pq+2q**2)

(9+3+4)=(3-3+2)(3+3+2)
16 = 2 * 8

9*1+3*1*4+4*16 = (3*1-3*1*2+2*(2*2))(3*1+3*1*2+2*(2**2))

85 = 5*17

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25p**4+4p**2 q**2+4q**4=(5p**2-4pq+2q**2)(5p**2+4pq+2q**2)

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16p**4 +8p**2 q**2 +9q**4=(4p**2-4pq+3q**2)(4p**2+4pq+3q**2)

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25p**4+5p**2 q**2+9q**4=(5p**2-5pq+3q**2)(5p**2+5pq+3q**2)

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These polynomials are all special cases of

A**2 p**4 + [2 A C - B**2] p**2 q**2 + C**2 q**4
= (A p**2 - B p q + C q**2)(A p**2 + B p q + C q**2)

Note that the special case, A = 2, B = 2, C = 1,
yields the form,
4 P**4 + q**4 = (2 p**2 - 2 pq + q**2)(2 p**2 + 2 pq + q**2)

which has application to factoring numbers of the form

2**(4k+2) + 1 = (2**(2k+1)-2**(k+1)+1)(2**(2k+1)-2**(k+1)+1)

2**6 + 1 = (2**3-2**2+1)(2**3+2**2+1)=5*13

2**10 + 1 =(2**5-2**3+1)(2**5+2**3+1)=25*41

2**14 + 1 =(2**7-2**4+1)(2**7+2**4+1)=113*145

2**18 + 1 =(2**9-2**5+1)(2**9+2**5+1)=481*545

More generally,
A = (2m+1)**2, C = 2,B= 2(2m+1)
yields,

[(2m+1)p]**4 + 4 q**4
=([(2m+1)p]**2-2(2m+1) pq +2 q**2)
([(2m+1)p]**2+2(2m+1) pq +2 q**2)

1+4 = 1 * 5
81+4=(9-6+2)*(9+6+2)=5*17
625+4=(25-10+2)(25+10+2)=17*37