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HomepageDiophantine Equations
Computer Search Results
Jim Cullen



Results of Computer Searches for Solutions to
Various Diophantine Equations


(10x)4 + (100y)2 + 1 = z2


In the table below, the results are sorted according to the value of x, and then according to the value of y. The entries highlighted in blue are those solutions belonging to the family defined by {x,y} = {n,50n4}. The rest are members of unidentified families. The value of y has been constrained to less than or equal to one million. The search was terminated at x=3773 due to lack of results. The second search, also terminated at x=3773 due to lack of results, extended the value of y to five million. The additional results obtained are highlighted in a magenta color, unless they are a member of the {n,50n4} family. One special search was conducted for values of x from 2 to 8, with the value of y less than or equal to 100 million with no new results. The x entries marked with a *** are those that have two known solutions. The x entries marked with a ::: are two or more consecutive values of x, not in the {x,y} = {n,50n4} family, with a known solution. One curiosity is that there are two solutions for x=7^1, one solution for x=7^2 (49), one solution for x=7^3 (343), but after this the pattern is broken; no solution for x=7^4 (2401) has yet been found, with y less than or equal to 1 billion.

xyz
280080001
34050405001
4128001280001
5312503125001
6648006480001
*** 7105001
*** 712005012005001
820480020480001
932805032805001
1050000050000001
1173205073205001
121036800103680001
131428050142805001
141920800192080001
152531250253125001
163276800327680001
174176050417605001
273180326249
31288342885001
494214485001
597036785001
661578594157860001
7032897632901249
*** 9215837415860001
*** 922172196217221249
982012981249
10049990050000001
1081864176186421249
114112861721249
14039802000001
14514252014406249
15512312812545001
15760975261025001
16017596017781249
18017706018000001
19017960818320001
1991309864131046249
20613026213700001
210677048080001
2282447448244800001
25313703815125001
26137716038326249
27276645077001249
2751041320104406249
2781022684102560001
28095759696080001
28711819214406249
::: 3121212900121680001
::: 3133032668303425001
3291168058117306249
3401581794158601249
34373043673985001
36143202245125001
36322789826325001
3791101214406249
3971864814187146249
4012053564205985001
41344101447285001
45325723832906249
46719902229525001
48480689684021249
5161018782105300001
5214436554444485001
53325845238406249
53893503897881249
5857069234945001
5891256240130326249
*** 599965752103025001
xyz
*** 5993118678313925001
6301205124126880001
6743459514348921249
6841123260121680001
7362630006268521249
7604571858460820001
76569849091125001
79139371873925001
8403874080393781249
8602917514300980001
8781384004158421249
9133354032345606249
9221843450203001249
9291057618136506249
950499718103161249
9802412904259701249
9842069946228521249
9873708432383425001
10134663906477546249
10312873990306426249
10403335006350601249
10492394520263526249
10824911614504921249
::: 11141003904159621249
::: 11153769330396906249
11224019472421200001
11364624100480080001
11792594322294325001
1202333578148281249
12144063594432260001
1240589168164661249
12452931486331605001
12558876157505001
12781073706195460001
13363043424352821249
1379607310199626249
13992651060329526249
1420492238207561249
1447708596221046249
14974875912536625001
1506712146237721249
15613359608415025001
15933960624470385001
16722136646351860001
1751578230312005001
1772725962322281249
1844512978343881249
18951470026388026249
194513756378305001
20592360350485226249
20814519342625925001
22641972132549200001
22923453582628681249
23662207804601760001
24262992574660260001
24582185798642500001
255631614653321249
25724160206781460001
26214088944799446249
26923702730813801249
27581278214771321249
2782936688779600001
*** 27931584030796005001
*** 27934690584910246249
29144765040973701249
30292218498943925001
315617366461011060001
338619875521163600001






x4 + y4 + 3 = z2


I've done one preliminary search for values of x and y less than or equal to 10,000. Of interest are equations of the form x4+y4+k=z2 where k is an odd integer. When possible solutions are inspected mod 8 or mod 16, certain values of k may be eliminated; specifically, -11, -10, -6, -5, -4, -3, 5, 6, 10, 11, 12, 13, and any of these values plus or minus 16n, where n is any integer. For k=+1 or +5, with the same constraints on {x,y} values, no solutions have been found ( many are known for k=-1 ). For k=+3 we have the table below ( though there were none found for k=-3 ), and for k=+7 one solution was found in the range; x=3 y=3 z=13.

xyz
2310
116825680758
xyz
32510741158302
646587734541650
4015891681113042






x4 + y4 + 1 = z2


Another search currently in progress. No solutions have been found for values of x and y less than or equal to 10 million. Inspection mod 20 shows that both x and y must be divisible by ten in any possible solution. All fourth powers mod 20 are in the set {0,1,5,16}. From this we deduce that the only possible values that x4 + y4 + 1 can have mod 20 are in the set {1,2,3,6,7,11,12,17}. All squares mod 20 must be in the set {0,1,4,5,9,16}. The only common residue between the two is 1 mod 20. This happens when both x4 and y4 are congruent to 0 mod 20, and this only happens when both x and y themselves are congruent to 0 mod 10. In other words, both x and y must be multiples of ten. Searching only multiples of 10 for x and y speeds up the search algorithm by a factor of 100. Using a sieve mod 9 on top of this gave another three-fold increase in speed. The current search algorithm significantly reduces program looping even more based on the regular pattern of candidate solutions mod 2340. This has increased the speed of the search algorithm by another three times, finishing the range of {x,y} from 5 million to 6 million in about a day. The plan is to allow the program to run until we eliminate any possible solutions with {x,y} <= 10 million. A restricted range search will follow, searching for solutions where x and y are more comparable in size.

Given that, in the equation x4 + y4 + 1 = z2, where x and y must be multiples of ten, we can say something about the form of z2. Let k be a natural number. In any solution then, z2 can have only one of two possible forms:

Case One:   z2 = [ 5000k ± 1 ]2


Case Two:   z2 = [ 1250 ( 2k - 1 ) + (-1)k ]2


Let's begin with Case One. To reduce our equations, let x = 10u and y = 10v, with u and v being natural numbers greater than zero. Our equation then can be rewritten as:

( 10u )4 + ( 10v )4 + 1 = z2


which is equivalent to:

104 ( u4 + v4 ) + 1 = z2


We'll substitute in our Case One formula for z2, which then works out to:

104 ( u4 + v4 ) + 1 = [ 5000k ± 1 ]2


We can expand the right side, giving:

104 ( u4 + v4 ) + 1 = 256k2 ± 103k + 1


which then reduces to:

u4 + v4 = 2500k2 ± k


For Case Two, we will simplify the math by dividing this case into two classes: k even and k odd, depending on whether k is an even or an odd integer. The resulting forms for z2 are then:

Case Two k EVEN:   z2 = [ 1250 ( 2k - 1 ) + 1 ]2


Case Two k ODD:   z2 = [ 1250 ( 2k - 1 ) - 1 ]2


which, when expanded and factored, are equivalent to:

Case Two k EVEN:   z2 = 5000 ( 2k - 1 ) ( 625k - 312 ) + 1


Case Two k ODD:   z2 = 5000 ( 2k - 1 ) ( 625k - 313 ) + 1


We can now reformulate Case Two by substituting in our formulas for z2 as follows:

Case Two k EVEN:   104 ( u4 + v4 ) + 1 = 5000 ( 2k - 1 ) ( 625k - 312 ) + 1


Case Two k ODD:   104 ( u4 + v4 ) + 1 = 5000 ( 2k - 1 ) ( 625k - 313 ) + 1


which then reduce to:

Case Two k EVEN:   u4 + v4 = 1/2 ( 2k - 1 ) ( 625k - 312 ) = 1/2 ( 1250k2 - 1249k + 312 )


Case Two k ODD:   u4 + v4 = 1/2 ( 2k - 1 ) ( 625k - 313 ) = 1/2 ( 1250k2 - 1251k + 313 )






x4 + y4 - 1 = z2


I've done one search for values of {x,y} less than or equal to 10,000.

xyz
5755
817296
1313239
17321064
22311076
27371551
28472344
31462324
44634416
46979644
47766184
6411112984
919912831
9819137724
10423958136
11115226184
113319102559
118705497220
14323960671
15210231046784
15725368591
193668447776
20823971656
257560320480
295485250807
305643423785
319794638596
401664469304
421317310069489
48916032580711
505515367943
560785691432
577624512304
57717683143504
593812747256
59921554657865
60822875243416
643905917465
65518493445615
659421717788391
66511431379265
83916332758031
94415532571176
xyz
95815672621396
99310901543980
100718293495559
1212503925433976
121718443708896
129725476701769
1327713250895896
134123936002169
137120894751511
140720424615836
147115683275216
155325526944936
155928438440169
156725667027316
163717694119959
1747368313903601
177230099583104
1799336911802561
1826521727420564
1857311510297785
1873353212958904
1874307310075204
2437333712620311
2657617838815084
2851432520395295
2996695949252504
3073504227119044
3073768859854976
3112382317532776
3223580935307151
3343425921305431
3455351817194940
3472556733253424
3739442324044791
3823524331132871
3866553734107436
3947898782254929
4009881979418041
4091445125934431
5257554941374649
53339901102072161
6087725364344471
6201622754608871
6371818378303001






x4 + y4 - 7 = z2


I've done one search for values of x less than or equal to 34,000 and y less than or equal to 100,000.

xyz
225
14806403
26478228485
13216042572875
700794798467
xyz
274620776431708035
3640556633694723
6700681044638372043
8198347201207350397
1277829474883927675







URL: http://members.bex.net/jtcullen515/math10.htm
© 2011: Jim Cullen - all rights reserved.
Contact the author with any questions or comments: Jim Cullen
Last Update: January 2, 2011.

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