a^4 – b^4 = c^4 – d^4 = e^4 – f^4

Posted on June 5, 2016 by benvitalis

 
 
a^4 \; - \; b^4 \; = \; c^4 \; - \; d^4 \; = \; e^4 \; - \; f^4
 

((a+b)/2) \,((a-b)/2) \,(((a+b)/2)^2+((a-b)/2)^2) \; = \; (a^4-b^4)/8
 

for example,
 

(1)

335084^4-296668^4=265076^4-93436^4=264047^4-1169^4=4860992489864937000960

(a+b)/2 = (335084+296668)/2 = 315876 …… (a-b)/2 = (335084-296668)/2 = 19208
(c+d)/2 = (265076+93436)/2 = 179256 ……. (c-d)/2 = (265076-93436)/2 = 85820
(e+f)/2 = (264047+1169)/2 = 132608 …….. (e-f)/2 = (264047-1169)/2 = 131439

19208 \; \times \; 315876 \; \times \; (19208^2 + 315876^2)
= \; 85820 \; \times \; 179256 \; \times \; (85820^2 + 179256^2)
= \; 131439 \; \times \; 132608 \; \times \; (131439^2 + 132608^2)
= \; 607624061233117125120
= \; 480 \; \times \; 1265883460902327344

 

(2)

421296^4-273588^4=415137^4-248289^4=401168^4-17228^4=25900232113758000049920

(421296+273588)/2 = 347442   …..   (421296-273588)/2 = 73854
(415137+248289)/2 = 331713   …..   (415137-248289)/2 = 83424
(401168+17228)/2 = 209198   ……   (401168-17228)/2 = 191970

347442 \; \times \; 73854 \; \times \; (347442^2 + 73854^2)
= \; 331713 \; \times \; 83424 \; \times \; (331713^2 + 83424^2)
= \; 209198 \; \times \; 191970 \; \times \; (209198^2 + 191970^2)
= \; 3237529014219750006240
= \; 480 \; \times \; 6744852112957812513

 

(3)

854688^4-813396^4=747633^4-682161^4=614656^4-465236^4=95886112914017928564480

(854688+813396)/2 = 834042   …..   (854688-813396)/2 = 20646
(747633+682161)/2 = 714897   …..   (747633-682161)/2 = 32736
(614656+465236)/2 = 539946   …..   (614656-465236)/2 = 74710

834042 \; \times \; 20646 \; \times \; (834042^2 + 20646^2)
= \; 714897 \; \times \; 32736 \; \times \; (714897^2 + 32736^2)
= \; 539946 \; \times \; 74710 \; \times \; (539946^2 + 74710^2)
= \; 11985764114252241070560
= \; 480 \; \times \; 24970341904692168897
 

In each of the 3 pairs   (a,b),   (c,d),   (e,f)   both numbers have the same parity.
There are 2 pairs of even numbers, and 1 pair of odd numbers
The equal products are divisible by 480.

 
 
Find other solutions.

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

About benvitalis

math grad - Interest: Number theory
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