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Diophantine Equations Computer Search Results Jim Cullen

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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.

x y z 2 800 80001 3 4050 405001 4 12800 1280001 5 31250 3125001 6 64800 6480001 *** 7 10 5001 *** 7 120050 12005001 8 204800 20480001 9 328050 32805001 10 500000 50000001 11 732050 73205001 12 1036800 103680001 13 1428050 142805001 14 1920800 192080001 15 2531250 253125001 16 3276800 327680001 17 4176050 417605001 27 3180 326249 31 28834 2885001 49 4214 485001 59 7036 785001 66 1578594 157860001 70 328976 32901249 *** 92 158374 15860001 *** 92 2172196 217221249 98 2012 981249 100 499900 50000001 108 1864176 186421249 114 11286 1721249 140 3980 2000001 145 142520 14406249 155 123128 12545001 157 609752 61025001 160 175960 17781249 180 177060 18000001 190 179608 18320001 199 1309864 131046249 206 130262 13700001 210 67704 8080001 228 2447448 244800001 253 137038 15125001 261 377160 38326249 272 766450 77001249 275 1041320 104406249 278 1022684 102560001 280 957596 96080001 287 118192 14406249 ::: 312 1212900 121680001 ::: 313 3032668 303425001 329 1168058 117306249 340 1581794 158601249 343 730436 73985001 361 432022 45125001 363 227898 26325001 379 11012 14406249 397 1864814 187146249 401 2053564 205985001 413 441014 47285001 453 257238 32906249 467 199022 29525001 484 806896 84021249 516 1018782 105300001 521 4436554 444485001 533 258452 38406249 538 935038 97881249 585 70692 34945001 589 1256240 130326249 *** 599 965752 103025001

x y z *** 599 3118678 313925001 630 1205124 126880001 674 3459514 348921249 684 1123260 121680001 736 2630006 268521249 760 4571858 460820001 765 698490 91125001 791 393718 73925001 840 3874080 393781249 860 2917514 300980001 878 1384004 158421249 913 3354032 345606249 922 1843450 203001249 929 1057618 136506249 950 499718 103161249 980 2412904 259701249 984 2069946 228521249 987 3708432 383425001 1013 4663906 477546249 1031 2873990 306426249 1040 3335006 350601249 1049 2394520 263526249 1082 4911614 504921249 ::: 1114 1003904 159621249 ::: 1115 3769330 396906249 1122 4019472 421200001 1136 4624100 480080001 1179 2594322 294325001 1202 333578 148281249 1214 4063594 432260001 1240 589168 164661249 1245 2931486 331605001 1255 8876 157505001 1278 1073706 195460001 1336 3043424 352821249 1379 607310 199626249 1399 2651060 329526249 1420 492238 207561249 1447 708596 221046249 1497 4875912 536625001 1506 712146 237721249 1561 3359608 415025001 1593 3960624 470385001 1672 2136646 351860001 1751 578230 312005001 1772 725962 322281249 1844 512978 343881249 1895 1470026 388026249 1945 13756 378305001 2059 2360350 485226249 2081 4519342 625925001 2264 1972132 549200001 2292 3453582 628681249 2366 2207804 601760001 2426 2992574 660260001 2458 2185798 642500001 2556 31614 653321249 2572 4160206 781460001 2621 4088944 799446249 2692 3702730 813801249 2758 1278214 771321249 2782 936688 779600001 *** 2793 1584030 796005001 *** 2793 4690584 910246249 2914 4765040 973701249 3029 2218498 943925001 3156 1736646 1011060001 3386 1987552 1163600001

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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.

x y z 2 3 10 116 825 680758

x y z 325 1074 1158302 646 5877 34541650 4015 8916 81113042

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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 )

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x⁴ + y⁴ - 1 = z²

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

x y z 5 7 55 8 17 296 13 13 239 17 32 1064 22 31 1076 27 37 1551 28 47 2344 31 46 2324 44 63 4416 46 97 9644 47 76 6184 64 111 12984 91 99 12831 98 191 37724 104 239 58136 111 152 26184 113 319 102559 118 705 497220 143 239 60671 152 1023 1046784 157 253 68591 193 668 447776 208 239 71656 257 560 320480 295 485 250807 305 643 423785 319 794 638596 401 664 469304 421 3173 10069489 489 1603 2580711 505 515 367943 560 785 691432 577 624 512304 577 1768 3143504 593 812 747256 599 2155 4657865 608 2287 5243416 643 905 917465 655 1849 3445615 659 4217 17788391 665 1143 1379265 839 1633 2758031 944 1553 2571176

x y z 958 1567 2621396 993 1090 1543980 1007 1829 3495559 1212 5039 25433976 1217 1844 3708896 1297 2547 6701769 1327 7132 50895896 1341 2393 6002169 1371 2089 4751511 1407 2042 4615836 1471 1568 3275216 1553 2552 6944936 1559 2843 8440169 1567 2566 7027316 1637 1769 4119959 1747 3683 13903601 1772 3009 9583104 1799 3369 11802561 1826 5217 27420564 1857 3115 10297785 1873 3532 12958904 1874 3073 10075204 2437 3337 12620311 2657 6178 38815084 2851 4325 20395295 2996 6959 49252504 3073 5042 27119044 3073 7688 59854976 3112 3823 17532776 3223 5809 35307151 3343 4259 21305431 3455 3518 17194940 3472 5567 33253424 3739 4423 24044791 3823 5243 31132871 3866 5537 34107436 3947 8987 82254929 4009 8819 79418041 4091 4451 25934431 5257 5549 41374649 5333 9901 102072161 6087 7253 64344471 6201 6227 54608871 6371 8183 78303001

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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.

x y z 2 2 5 14 80 6403 26 478 228485 132 1604 2572875 700 794 798467

x y z 2746 20776 431708035 3640 5566 33694723 6700 68104 4638372043 8198 34720 1207350397 12778 29474 883927675

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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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