Problems & Puzzles: Puzzles

Puzzle 47.- p⁴ = a⁴ + b⁴ + c⁴ + d⁴, {a, b, c, d}>0

At (29/03/99) Felice Russo wrote:

 “Is there any 4 positive integers, let say a, b, c, d such that: p⁴ = a⁴ + b⁴ + c⁴ + d⁴

A check done for the primes lower than 3000 highligths only the number 353 (that is also a palprime): 353⁴ = 30⁴ + 120⁴ + 272⁴ + 315⁴

So the question is: How many other primes and palprimes of this form exist?”

Comment:

I (CR) have posted this problem not only because the solution (re)discovered by Russo contains a prime (a palprime, indeed!!), but because it conveys to a beauty story  of the mathematics of this kind of equations (Diophantine).

It happens that Euler (1707-1783) – probably generalizing the Fermat Last Theorem – believed that a cube was a sum of no less than 3 cubes, a biquadrate was a sum of no less than four biquadrates, and so on… . By the way, at the time he stated his conjecture there was not known any example of a biquadrate as a sum of four biquadrates. Not until at 1911 R. Norrie discovered the first example… the same now (re)discovered by Russo!!

Regarding the Euler’s mentioned conjecture, at the end it resulted to be false because five years after the Norrie’s result, Lander & Parkin got this:

144⁵ = 27⁵ + 84⁵ + 110⁵ + 133⁵!!!

Very recently (1988) Noam Elkies got:

2682440⁴ + 15365639⁴ + 18796760⁴ = 20615673⁴

and very soon, Roger Frye got

95800⁴ + 217519⁴ + 414560⁴ = 422481⁴

(See the whole story at “Unsolved Problems in Number Theory”, D1, pp. 139-144, by Richard K. Guy)

But… (Always a but!)… the Russo’s question remains, at least for me because I don't know any other solution (prime or composite) to p^4 = a^4 + b^4 + c^4 + d^4… so the game continues…

Warut Roonguthai sent (10/6/99) two solutions to Puzzle 47, previously reported in: http://www.astro.virginia.edu/~eww6n/math/DiophantineEquation4thPowers.html:

5281^4 = 1000^4+1120^4+3233^4+5080^4
7703^4 = 2230^4+3196^4+5620^4+6995^4

5281 & 7703 are primes...

Is there any other solution being p palprime?

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Jaroslaw Wroblewski wrote (22/6/6)

I have made complete list of all 438 solutions to z^4=a^4+b^4+c^4+d^4
with z<100,000. Attached is the list of all 75 solutions with z prime. In 3 cases, marked with !'s, z is a palprime. In 2 cases, marked with +'s, a is a prime.

1 {353, 272, 315, 120, 30}
!!!!!!!!!!
2 {5281, 3233, 5080, 1120, 1000}
3 {7703, 3196, 6995, 5620, 2230}
4 {9137, 816, 3285, 8570, 6180}
5 {9431, 5192, 6935, 7820, 5800}
6 {10939, 5936, 9145, 8530, 5300}
7 {11681, 3556, 8635, 10490, 5300}
8 {14029, 8278, 1275, 13410, 6420}
9 {14489, 9498, 13355, 8010, 1530}
10 {17519, 15644, 8495, 13060, 1220}
11 {17881, 7256, 1945, 17320, 9860}
12 {18077, 9423, 17510, 8340, 2760}
13 {18701, 12668, 15365, 12430, 11410}
14 {19483, 15906, 8355, 16560, 610}
15 {20719, 19142, 8855, 13970, 8720}
16 {21013, 3362, 16525, 17740, 12070}
17 {22247, 21046, 11485, 12860, 7960}
18 {22961, 777, 22950, 4800, 1010}
19 {24953, 20196, 21585, 7690, 5250}
20 {25589, 302, 4525, 25540, 7310}
21 {25913, 24712, 9795, 14940, 11730}
22 {26821, 18297, 20610, 19380, 17020}
23 {26987, 23638, 14025, 20580, 3420}
24 {27893, 3518, 425, 27340, 14680}
25 {28297, 4321, 27570, 15840, 3090}
26 {34327, 19327, 33430, 1880, 1090}
27 {37273, 28189, 30040, 26380, 2750}
!!!!!!!!!!
28 {39671, 16128, 7795, 38100, 23370}
29 {40031, 39344, 8005, 19660, 11630}
30 {40129, 19728, 38235, 23490, 2720}
31 {42209, 33913, 31280, 30590, 11650}
32 {42359, 15484, 8695, 41140, 23240}
33 {43397, 18621, 35670, 34170, 25820}
34 {43781, 26844, 37625, 32010, 17820}
35 {46679, 1054, 485, 39490, 39010}
36 {50587, 48788, 15385, 29720, 14710}
37 {51361, 31764, 27385, 48150, 7590}
38 {51659, 43188, 20555, 40080, 30660}
39 {52183, 38556, 46675, 25980, 7710}
40 {52253, 28372, 50575, 22630, 6860}
41 {56857, 50474, 16945, 40940, 32150}
42 {57503, 9378, 29625, 56450, 5820}
43 {60631, 43744, 20695, 55760, 6520}
44 {61979, 36771, 51660, 48290, 24630}
45 {62303, 11678, 55815, 45510, 32040}
46 {62497, 49378, 43015, 48790, 21680}
47 {63127, 42502, 5825, 57800, 34730}
48 {67129, 52522, 34315, 54850, 38770}
49 {70121, 27022, 15935, 59260, 57910}
50 {70321, 5321, 59350, 54500, 42370}
51 {72923, 32611, 65410, 54530, 4760}
++++++++++
52 {74293, 30701, 65800, 57250, 17210}
53 {76283, 15869, 76090, 22960, 2950}
54 {77003, 70796, 42025, 51160, 16150}
55 {77377, 9252, 20955, 65010, 64940}
!!!!!!!!!!
56 {77557, 15682, 16745, 71540, 56020}
57 {79337, 33166, 21505, 72310, 57400}
58 {79967, 62256, 68805, 42950, 15360}
59 {80701, 9918, 22185, 73330, 60330}
60 {81157, 30532, 77635, 49850, 10010}
61 {81619, 79744, 18305, 44170, 12020}
62 {81973, 55914, 16065, 75250, 42450}
63 {82387, 9434, 75265, 60700, 25120}
64 {82613, 36762, 67885, 69570, 17340}
65 {83617, 64206, 64455, 60370, 34080}
66 {85889, 30361, 75650, 67250, 24560}
67 {86399, 51493, 75560, 63340, 4450}
68 {87071, 41922, 34745, 74550, 68520}
69 {90373, 71502, 57795, 73620, 13550}
70 {90379, 46478, 46445, 85610, 43820}
71 {91237, 69987, 74460, 61610, 19770}
72 {96013, 7888, 81595, 79630, 25810}
73 {97429, 58679, 93730, 29240, 24100}
++++++++++
74 {98017, 80517, 81600, 45110, 36600}
75 {98429, 65297, 91420, 48940, 17650}

Later he added:

I have extended the search of
z^4=a^4+b^4+c^4+d^4 up to 200,000.

There are 465 primitive solutions with z between 100,000 and 200,000
including 88 solutions with z prime, but no palprimes.

The following solutions may be of some interest:

154789 49586 145615 102170 55450
154789 121664 66405 119760 106740

195479 112822 189695 41080 25570
195479 140947 180520 43340 12970

Those are prime z's which allow 2 solutions.
There are also 3 z's below 100,000, which allow 2 primitive solutions,
but none of them is prime.

Therefore the equation
P^4=a^4+b^4+c^4+d^4=e^4+f^4+g^4+h^4
with P prime has exactly 2 nontrivial solutions with P<200,000.

 

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