As previously conjectured:

Where p is prime, 1 < a

_{1}< a

_{2}+...+ < a

_{m}< p, if there exists

{n, a

_{1}, a

_{2}, ..., a

_{m}, p, x} that can satisfy (a

_{1}

^{n}+ a

_{2}

^{n}+...+ a

_{m}

^{n}) / p

^{n-1}= x, then there exists, with the same number of terms (a) and exponent (n), {n, a

_{1}, a

_{2}, ... , a

_{m}, b} that can satisfy a

_{1}

^{n}+ a

_{2}

^{n}+...+ a

_{m}

^{n}= b

^{n}.

An example of a {n, a

_{1}, a

_{2}, a

_{3}, a

_{4}, p, x} that satisfies

(a

_{1}

^{5}+ a

_{2}

^{5}+ a

_{3}

^{5}+ a

_{4}

^{5}) / p

^{4}= x, is {5, 31, 40, 43, 51, 61, 45}.

(31

^{5}+ 40

^{5}+ 43

^{5}+ 51

^{5}) / 61

^{4}= 45

Therefore, there exists {n, a

_{1}, a

_{2}, a

_{3}, a

_{4}, b} that can satisfy

a

_{1}

^{5}+ a

_{2}

^{5}+ a

_{3}

^{5}+ a

_{4}

^{5}= b

^{5}.

Suppose that if there are solutions to either equation, then there are solutions to both. And that all solutions to either type correlate to solutions of the other.

The example in the conjecture is a 5th power. Therefore, I'll be covering 5th powers first, then 4th powers, and 6th powers.

**5th Powers**

The lowest 5th power solution having the sum of 4 terms that equal one term is

27

^{5}+ 84

^{5}+ 110

^{5}+ 133

^{5}= 144

^{5}(Lander and Parkin, 1967)

and has a modular correlation with

(31

^{5}+ 40

^{5}+ 43

^{5}+ 51

^{5}) / 61

^{4}= 45

144 - 40 - 43 = 0 mod 61

133 - 31 = 0 mod 51

110 + 61 -51 = 0 mod 40

84 + 45 = 0 mod 43

27 + 61 - 43 = 0 mod 45

The only other solution currently known is

55

^{5}+ 3183

^{5}+ 28969

^{5}+ 85282

^{5}= 85359

^{5}(R. Frye 2004)

and correlates with

(34

^{5}+ 179

^{5}+ 185

^{5}+ 222

^{5}) / 223

^{4}= 380

85359 - 223 = 0 mod 34

85282 + 222 - 34 = 0 mod 185

28969 + 222 + 179 - 380 = 0 mod 223

3183 + 34 + 185 + 222 - 223 = 0 mod 179

55 - 34 - 179 - 222 = 0 mod 380

There are a total of 9 equations where p <= 1249.

(31

^{5}+ 40

^{5}+ 43

^{5}+ 51

^{5}) / 61

^{4}= 45

(34

^{5}+ 179

^{5}+ 185

^{5}+ 222

^{5}) / 223

^{4}= 380

(67

^{5}+ 129

^{5}+ 240

^{5}+ 292

^{5}) / 313

^{4}= 308

(20

^{5}+ 23

^{5}+ 312

^{5}+ 343

^{5}) / 373

^{4}= 398

(110

^{5}+ 234

^{5}+ 529

^{5}+ 709

^{5}) / 739

^{4}= 742

(36

^{5}+ 375

^{5}+ 393

^{5}+ 719

^{5}) / 761

^{4}= 623

(7

^{5}+ 304

^{5}+ 434

^{5}+ 461

^{5}) / 1019

^{4}= 36

(14

^{5}+ 608

^{5}+ 868

^{5}+ 922

^{5}) / 1019

^{4}= 1152

(116

^{5}+ 639

^{5}+ 706

^{5}+ 980

^{5}) / 1153

^{4}= 671

Each of the 9 equations can correlate with multiple 5th power solutions. Additional 5th power solutions exist that have yet to be discovered.

**4th Powers**

The first 4th power solution discovered having the sum of 3 terms that equal one term is

2682440

^{4}+ 15365639

^{4}+ 18796760

^{4}= 20615673

^{4}(N. Elkies 1986)

and correlates with

(24

^{4}+ 87

^{4}+ 109

^{4}) / 197

^{3}= 26

20615673 + 24 + 97 = 0 mod 26

18796760 + 24 + 26 - 87 = 0 mod 109

15365639 + 24 + 197 = 0 mod 87

2682440 + 109 = 0 mod 197

The lowest 4th power solution is

95800

^{4}+ 217519

^{4}+ 414560

^{4}= 422481

^{4}(R. Frye 1988)

also correlates with

(24

^{4}+ 87

^{4}+ 109

^{4}) / 197

^{3}= 26

422481 + 197 + 26 - 87 - 24 = 0 mod 109

414560 - 87 = 0 mod 197

217519 - 197 - 26 = 0 mod 24

95800 + 197 - 109 = 0 mod 26

There are a total of 25 equations where p <= 1249.

(24

^{4}+ 87

^{4}+ 109

^{4}) / 197

^{3}= 26

(51

^{4}+ 175

^{4}+ 197

^{4}) / 223

^{3}= 221

(2

^{4}+ 107

^{4}+ 330

^{4}) / 419

^{3}= 163

(64

^{4}+ 319

^{4}+ 351

^{4}) / 439

^{3}= 302

(9

^{4}+ 40

^{4}+ 318

^{4}) / 449

^{3}= 113

(70

^{4}+ 255

^{4}+ 348

^{4}) / 449

^{3}= 209

(10

^{4}+ 301

^{4}+ 398

^{4}) / 479

^{3}= 303

(80

^{4}+ 223

^{4}+ 341

^{4}) / 503

^{3}= 126

(71

^{4}+ 322

^{4}+ 415

^{4}) / 557

^{3}= 234

(43

^{4}+ 387

^{4}+ 575

^{4}) / 599

^{3}= 613

(100

^{4}+ 155

^{4}+ 536

^{4}) / 673

^{3}= 273

(24

^{4}+ 479

^{4}+ 587

^{4}) / 727

^{3}= 446

(209

^{4}+ 211

^{4}+ 519

^{4}) / 797

^{3}= 151

(638

^{4}+ 701

^{4}+ 722

^{4}) / 797

^{3}= 1341

(374

^{4}+ 469

^{4}+ 544

^{4}) / 823

^{3}= 279

(128

^{4}+ 150

^{4}+ 627

^{4}) / 839

^{3}= 263

(105

^{4}+ 236

^{4}+ 588

^{4}) / 863

^{3}= 191

(8

^{4}+ 190

^{4}+ 881

^{4}) / 929

^{3}= 753

(120

^{4}+ 653

^{4}+ 849

^{4}) / 941

^{3}= 842

(91

^{4}+ 822

^{4}+ 906

^{4}) / 947

^{3}= 1331

(387

^{4}+ 760

^{4}+ 797

^{4}) / 1049

^{3}= 658

(245

^{4}+ 591

^{4}+ 1045

^{4}) / 1069

^{3}= 1079

(189

^{4}+ 635

^{4}+ 738

^{4}) / 1151

^{3}= 302

(195

^{4}+ 661

^{4}+ 917

^{4}) / 1217

^{3}= 499

(167

^{4}+ 488

^{4}+ 923

^{4}) / 1249

^{3}= 402

Many 4th power solutions have been discovered. Certainly there are many more.

**6th Powers**

Sum of six terms of the sixth power:

(9

^{6}+ 15

^{6}+ 19

^{6}+ 20

^{6}+ 28

^{6}+ 34

^{6}) / 37

^{5}= 31 (John Y, Project Primality, 2009)

(38

^{6}+ 49

^{6}+ 94

^{6}+ 130

^{6}+ 137

^{6}+ 147

^{6}) / 179

^{5}= 121 (John Y, Project Primality, 2011)

Indicates that solutions exist for a

_{1}

^{6}+ a

_{2}

^{6}+ a

_{3}

^{6}+ a

_{4}

^{6}+ a

_{5}

^{6}+ a

_{6}

^{6}= b

^{6}. As of this writing none have been discovered.

Sum of five terms of the sixth power:

(278

^{6}+ 373

^{6}+ 798

^{6}+ 954

^{6}+ 1004

^{6}) / 1303

^{5}= 543 (John Y, Project Primality, 2011)

(58

^{6}+ 212

^{6}+ 577

^{6}+ 893

^{6}+ 978

^{6}) / 1439

^{5}= 230 (John Y, Project Primality, 2011)

These are the only two examples where p<=1439.

The integers may be large enough to assist in finding a solution for

a

_{1}

^{6}+ a

_{2}

^{6}+ a

_{3}

^{6}+ a

_{4}

^{6}+ a

_{5}

^{6}= b

^{6}.

Contact Info: projectprimality@yahoo.comLast update: May 12, 2012