Chapter 4 / 5 combined

Concise version - results only

e(1,3,2) = 1 : 2³ = 3² - 1

e(1,5,2) £ 3 : 1⁵ = 2² - 3, else 2⁵ = 6² - 4

e(1,7,2) £ 7 : 2⁷ = 11² + 7

e(2,2,2) = 0 : 3² + 4² = 5² (Pythagoras)

e(2,3,2) = 0 : 2³ + 1³ = 3² (smallest), 9265³ + 2184³ = 897623² (largest)

e*(2,3,2) = 0 : 8³ - 7³ = 13² (smallest), 9632³ - 2743³ = 934331² (largest)

e(2,4,2) = 1 : 7⁴ + 5⁴ = 55² + 1 (smallest), 9901⁴ + 5333⁴ = 102072161² + 1 (largest)

e*(2,4,2) = 1 : 3⁴ - 2⁴ = 8² + 1 (smallest), 9961⁴ - 2873⁴ = 98877599² - 1 (largest)

e(2,5,2) £ 2 : 3⁵ + 3⁵ = 22² + 2 or 111⁵ + 87⁵ = 147766² + 2

e*(2,5,2) £ 1 : 70⁵ - 56⁵ = 33615² - 1

e(2,6,2) £ 1 : 31⁶ + 29⁶ = 38501² + 1

e*(2,6,2) £ 1 : 16⁶ - 14⁶ = 3041² - 1, 78⁶ - 23⁶ = 474396² - 1

e(2,7,2) £ 7 : 4⁷ + 4⁷ = 181² + 7

e*(2,7,2) £ 1 : 64⁷ - 63⁷ = 677544² + 1

e(3,3,2) = 0 : 3³ + 2³ + 1³ = 6² (smallest), 14³ + 9³ + 2³ = 59², 25³ + 24³ + 23³ = 204²

e*(3,3,2) = 0 : 3³ - 1³ - 1³ = 5² and 5³ + 3³ - 2³ = 12²

e(3,4,2) = 0 : 20⁴ + 15⁴ + 12⁴ = 481²

e*(3,4,2) = 0 : 3⁴ - 2⁴ - 1⁴ = 8², 7⁴ + 5⁴ - 1⁴ = 55² and 8⁴ + 7⁴ - 4⁴ = 79²

e(3,5,2) = 0 : 69⁵ + 49⁵ + 3⁵ = 42971², 81⁵ + 68⁵ + 20⁵ = 70313²

e*(3,5,2) = 0 : 15⁵ + 5⁵ - 14⁵ = 474², 16⁵ + 3⁵ - 15⁵ = 538² and 58⁵ + 21⁵ - 33⁵ = 24926²

e(3,6,2) = 0 : 100⁶ + 81⁶ + 42⁶ = 1134865²

e*(3,6,2) = 0 : 4⁶ + 3⁶ - 2⁶ = 69² (smallest), 57⁶ - 44⁶ - 28⁶ = 162967²

e(3,7,2) £ 1 : 4⁷ + 2⁷ + 2⁷ = 129² - 1 (parameterised solution), else 5⁷ + 4⁷ + 4⁷ = 333² + 4

e*(3,7,2) = 0 : 16⁷ - 9⁷ - 7⁷ = 16212² and 64⁷ - 63⁷ - 1⁷ = 677544²

e(4,7,2) = 0 : 4⁷ + 2⁷ + 2⁷ + 1⁷ = 129², 74⁷ + 59⁷ + 17⁷ + 4⁷ = 3826270², 78⁷ + 51⁷ + 42⁷ + 25⁷ = 4324306²

e(1,5,3) £ 5 : 2⁵ = 3³ + 5

e(1,7,3) £ 3 : 2⁷ = 5³ + 3

e(2,2,3) = 0 : 11² + 2² = 5³ (smallest), 9830² + 1159² = 461³ (largest)

e*(2,2,3) = 0 : 3² - 1² = 2³ (smallest)

e(2,3,3) = 1 : 10³ + 9³ = 12³ + 1 and 8³ + 6³ = 9³ - 1 (smallest)

e(2,4,3) £ 5 : 2⁴ + 2⁴ = 3³ + 5

e*(2,4,3) £ 1 : 3⁴ - 2⁴ = 4³ + 1

e(2,5,3) £ 7 : 5⁵ + 3⁵ = 15³ - 7

e*(2,5,3) £ 4 : 2⁵ - 1⁵ = 3³ + 4, else 3⁵ - 2⁵ = 6³ - 5

e(2,6,3) £ 3 : 2⁶ + 2⁶ = 5³ + 3

e*(2,6,3) : no improvement : 4⁶ - 3⁶ = 15³ - 8

e(2,7,3) £ 4 : 2⁷ + 1⁷ = 5³ + 4, else 3⁷ + 1⁷ = 13³ - 9

e*(2,7,3) £ 2 : 2⁷ - 1⁷ = 5³ + 2

e(3,3,3) = 0 : 5³ + 4³ + 3³ = 6³ (smallest)

e(3,4,3) = 0 : 5⁴ + 5⁴ + 3⁴ = 11³

e*(3,4,3) = 0 : 3⁴ - 2⁴ - 1⁴ = 4³, 9⁴ - 8⁴ - 7⁴ = 4³ and 31⁴ + 20⁴ - 17⁴ = 100³ (largest)

e(3,5,3) £ 3 : 81⁵ + 74⁵ + 15⁵ = 1787³ - 3

e*(3,5,3) £ 1 : 92⁵ + 33⁵ - 74⁵ = 1640³ + 1 and 69⁵ - 67⁵ - 9⁵ = 598³ + 1

e(3,6,3) £ 2 : 25⁶ + 23⁶ + 6⁶ = 732³ + 2

e*(3,6,3) = 0 : 19⁶ - 18⁶ - 15⁶ = 118³

e(3,7,3) £ 8 : 3⁷ + 1⁷ + 1⁷ = 13³ - 8, else 15⁷ + 5⁷ + 4⁷ = 555³ + 9

e*(3,7,3) £ 1 : 2⁷ - 1⁷ - 1⁷ = 5³ + 1

e(4,5,3) = 0 : 4⁵ + 3⁵ + 2⁵ + 2⁵ = 11³ (smallest), 48⁵ + 8⁵ + 5⁵ + 3⁵ = 634³ (largest)

e*(4,5,3) = 0 : 22⁵ + 19⁵ + 13⁵ - 4⁵ = 200³ and 25⁵ + 25⁵ + 13⁵ - 2⁵ = 271³

e(4,6,3) £ 3 : 57⁶ + 19⁶ + 19⁶ + 10⁶ = 3252³ + 3 and 68⁶ + 53⁶ + 25⁶ + 15⁶ = 4950³ + 3

e*(4,6,3) £ 1 : 8⁶ - 5⁶ - 4⁶ - 4⁶ = 62³ - 1 and 6⁶ + 2⁶ - 5⁶ - 4⁶ = 30³ - 1

e(4,7,3) £ 7 : 3⁷ + 1⁷ + 1⁷ + 1⁷ = 13³ - 7, else 10⁷ + 6⁷ + 5⁷ + 3⁷ = 218³ + 16

e*(4,7,3) = 0 : 2⁷ - 1⁷ - 1⁷ - 1⁷ = 5³, else 29⁷ + 23⁷ + 11⁷ - 15⁷ = 2737³ - 1

e(5,6,3) £ 2 : 28⁶ + 21⁶ + 15⁶ + 10⁶ + 6⁶ = 834³ + 2 and 28⁶ + 21⁶ + 16⁶ + 16⁶ + 3⁶ = 844³ + 2

e(5,7,3) £ 6 : 3⁷ + 1⁷ + 1⁷ + 1⁷ + 1⁷ = 13³ - 6

e(6,6,3) £ 3 : 10⁶ + 9⁶ + 7⁶ + 7⁶ + 4⁶ + 3⁶ = 121³ + 3 (smallest)

e(6,7,3) £ 1 : 20⁷ + 7⁷ + 2⁷ + 2⁷ + 2⁷ + 2⁷ = 1086³ - 1

e(7,6,3) = 0 : 30⁶ + 24⁶ + 9⁶ + 9⁶ + 3⁶ + 3⁶ + 1⁶ = 973³

e(7,7,3) = 0 : 20⁷ + 7⁷ + 2⁷ + 2⁷ + 2⁷ + 2⁷ + 1⁷ = 1086³

e(1,3,4) £ 11 : 3³ = 2⁴ + 11

e(1,5,4) £ 13 : 3⁵ = 4⁴ - 13

e(1,7,4) £ 47 : 2⁷ = 3⁴ + 47

e(2,2,4) = 0 : 24² + 7² = 5⁴, 120² + 119² = 13⁴ and 240² + 161² = 17⁴ (from Pythagoras)

e*(2,2,4) = 0 : 5² - 3² = 2⁴ (smallest)

e(2,3,4) £ 3 : 63³ + 31³ = 23⁴ - 3

e*(2,3,4) £ 1 : 21³ - 19³ = 7⁴ + 1 and 37³ - 3³ = 15⁴ + 1

e(2,4,4) £ 46 : 5⁴ + 5⁴ = 6⁴ - 46

e(2,5,4) £ 12 : 3⁵ + 1⁵ = 4⁴ - 12, else 2⁵ + 2⁵ = 3⁴ - 17

e*(2,5,4) : no improvement

e(2,6,4) £ 16 : 2⁶ + 1⁶ = 3⁴ - 16

e*(2,6,4) : no improvement

e(2,7,4) £ 86 : 3⁷ + 2⁷ = 7⁴ - 86

e*(2,7,4) £ 33 : 6⁷ - 2⁷ = 23⁴ - 33

e(3,3,4) = 0 : 20³ + 17³ + 12³ = 11⁴ and 74³ + 48³ + 25³ = 27⁴

e*(3,3,4) = 0 : 15³ + 9³ - 2³ = 8⁴ and 11³ - 3³ - 2³ = 6⁴

e(3,4,4) = 0 : 414560⁴ + 217519⁴ + 95800⁴ = 422481⁴ (Frye), else 47⁴ + 22⁴ + 21⁴ = 48⁴ + 2

e(3,5,4) £ 3 : 4⁵ + 3⁵ + 2⁵ = 6⁴ + 3

e*(3,5,4) : no improvement : 16⁵ + 13⁵ - 17⁵ = 2⁴ - 4

e(3,6,4) £ 15 : 2⁶ + 1⁶ + 1⁶ = 3⁴ - 15

e*(3,6,4) : no improvement

e(3,7,4) £ 22 : 4⁷ + 3⁷ + 3⁷ = 12⁴ + 22

e*(3,7,4) : no improvement

e(4,5,4) £ 3 : 104⁵ + 72⁵ + 52⁵ + 45⁵ = 348⁴ - 3

e*(4,5,4) £ 1 : 9⁵ + 5⁵ - 7⁵ - 7⁵ = 13⁴ - 1, 16⁵ + 13⁵ + 3⁵ - 17⁵ = 4⁴ - 1

e(4,6,4) £ 10 : 21⁶ + 18⁶ + 11⁶ + 3⁶ = 105⁴ + 10

e*(4,6,4) £ 1 : 6⁶ + 4⁶ - 2⁶ - 2⁶ = 15⁴ - 1

e(4,7,4) £ 16 : 9⁷ + 5⁷ + 4⁷ + 3⁷ = 47⁴ - 16

e*(4,7,4) £ 5 : 29⁷ + 10⁷ + 6⁷ - 25⁷ = 325⁴ - 5

e(5,5,4) = 0 : 185⁵ + 169⁵ + 109⁵ + 46⁵ + 1⁵ = 780⁴

e(5,6,4) £ 3 : 9⁶ + 8⁶ + 5⁶ + 3⁶ + 2⁶ = 30⁴ + 3

e(5,7,4) £ 7 : 31⁷ + 14⁷ + 13⁷ + 11⁷ + 10⁷ = 408⁴ + 7

e(6,6,4) £ 1 : 9⁶ + 8⁶ + 7⁶ + 4⁶ + 4⁶ + 4⁶ = 31⁴ + 1

e(6,7,4) £ 8 : 31⁷ + 14⁷ + 13⁷ + 11⁷ + 10⁷ + 1⁷ = 408⁴ + 8, else 16⁷ + 13⁷ + 10⁷ + 7⁷ + 5⁷ + 4⁷ = 136⁴ + 9

e(7,6,4) = 0 : 92⁶ + 66⁶ + 57⁶ + 50⁶ + 48⁶ + 22⁶ + 8⁶ = 931⁴

e(1,7,5) £ 115 : 2⁷ = 3⁵ - 115

e(2,2,5) = 0 : 41² + 38² = 5⁵ (smallest), 6121² + 5646² = 37⁵ (largest)

e*(2,2,5) = 0 : 9² - 7² = 2⁵

e(2,3,5) £ 2 : 271³ + 239³ = 32⁵ - 2

e*(2,3,5) £ 2 : 37³ - 35³ = 6⁵ + 2

e(2,4,5) £ 65 : 22⁴ + 11⁴ = 12⁵ + 65

e*(2,4,5) £ 1 : 37⁴ - 34⁴ = 14⁵ + 1

e(2,5,5) £ 12 : 16⁵ + 13⁵ = 17⁵ + 12

e(2,6,5) £ 115 : 2⁶ + 2⁶ = 3⁵ - 115

e*(2,6,5) : no improvement

e(2,7,5) £ 13 : 2⁷ + 2⁷ = 3⁵ + 13

e*(2,7,5) : no improvement

e(3,3,5) = 0 : 77³ + 57³ + 49³ = 15⁵ and 86³ + 40³ + 39³ = 15⁵

e*(3,3,5) = 0 : 17³ + 15³ - 8³ = 6⁵, 41³ - 33³ - 6³ = 8⁵ and 48³ - 21³ - 11³ = 10⁵

e(3,4,5) £ 1 : 80⁴ + 66⁴ + 27⁴ = 36⁵ + 1

e*(3,4,5) = 0 : 7⁴ - 6⁴ - 3⁴ = 4⁵ and 37⁴ - 34⁴ - 1⁴ = 14⁵

e(3,5,5) £ 18 : 13⁵ + 13⁵ + 7⁵ = 15⁵ + 18

e(3,6,5) £ 51 : 2⁶ + 2⁶ + 2⁶ = 3⁵ - 51

e*(3,6,5) £ 9 : 31⁶ - 21⁶ - 17⁶ = 60⁵ - 9

e(3,7,5) £ 14 : 2⁷ + 2⁷ + 1⁷ = 3⁵ + 14, else 2⁷ + 1⁷ + 1⁷ = 3⁵ - 113

e*(3,7,5) £ 1 : 19⁷ + 7⁷ - 18⁷ = 49⁵ + 1

e(4,4,5) = 0 : 33⁴ + 33⁴ + 17⁴ + 12⁴ = 19⁵ and 37⁴ + 26⁴ + 19⁴ + 11⁴ = 19⁵

e(4,5,5) = 0 : 133⁵ + 110⁵ + 84⁵ + 27⁵ = 144⁵ (Lander & Parkin)

e(4,6,5) £ 13 : 2⁶ + 2⁶ + 2⁶ + 2⁶ = 3⁵ + 13

e*(4,6,5) £ 8 : 31⁶ + 1⁶ - 21⁶ - 17⁶ = 60⁵ - 8

e(4,7,5) £ 15 : 2⁷ + 2⁷ + 1⁷ + 1⁷ = 3⁵ + 15, else 4⁷ + 2⁷ + 2⁷ + 2⁷ = 7⁵ - 39

e*(4,7,5) = 0 : 19⁷ + 7⁷ + 18⁷ - 1⁷ = 49⁵ (from k = 3)

e(5,6,5) £ 4 : 5⁶ + 5⁶ + 3⁶ + 3⁶ + 2⁶ = 8⁵ + 4

e(5,7,5) £ 16 : 2⁷ + 2⁷ + 1⁷ + 1⁷ + 1⁷ = 3⁵ + 16, else 4⁷ + 2⁷ + 2⁷ + 2⁷ + 1⁷ = 7⁵ - 38

e(6,6,5) £ 5 : 5⁶ + 5⁶ + 3⁶ + 3⁶ + 2⁶ + 1⁶ = 8⁵ + 5, else 8⁶ + 6⁶ + 5⁶ + 5⁶ + 5⁶ + 5⁶ = 13⁵ + 7

e(6,7,5) £ 17 : 2⁷ + 2⁷ + 1⁷ + 1⁷ + 1⁷ + 1⁷ = 3⁵ + 17, else 4⁷ + 2⁷ + 2⁷ + 2⁷ + 1⁷ + 1⁷ = 7⁵ - 37

e(7,6,5) = 0 : 90⁶ + 85⁶ + 68⁶ + 68⁶ + 57⁶ + 54⁶ + 19⁶ = 259⁵

e(7,7,5) = 0 : 46⁷ + 45⁷ + 32⁷ + 23⁷ + 12⁷ + 7⁷ + 6⁷ = 243⁵

e(1,7,6) £ 759 : 4⁷ = 5⁶ + 759

e(2,2,6) = 0 : 117² + 44² = 5⁶ , 2035² + 828² = 13⁶ and 4888² + 495² = 17⁶ (from n = 3)

e*(2,2,6) = 0 : 17² - 15² = 2⁶

e(2,3,6) = 1 : 8³ + 6³ = 3⁶ - 1

e*(2,3,6) = 1 : 12³ - 10³ = 3⁶ - 1

e(2,4,6) £ 23 : 5⁴ + 3⁴ = 3⁶ - 23

e*(2,4,6) £ 1 : 3⁴ - 2⁴ = 2⁶ + 1

e(2,5,6) £ 53 : 5⁵ + 4⁵ = 4⁶ + 53

e*(2,5,6) £ 52 : 4⁵ - 3⁵ = 3⁶ + 52

e(2,6,6) £ 601 : 2⁶ + 2⁶ = 3⁶ - 601, else 8⁶ + 8⁶ = 9⁶ - 7153

e(2,7,6) £ 278 : 3⁷ + 3⁷ = 4⁶ + 278

e*(2,7,6) : no improvement

e(3,3,6) = 0 : 8³ + 6³ + 1³ = 3⁶ and 22³ + 17³ + 4³ = 5⁶ (from n = 3)

e*(3,3,6) = 0 : 12³ + 1³ - 10³ = 3⁶, 15³ + 9³ - 2³ = 4⁶ and 6³ - 5³ - 3³ = 2⁶

e(3,4,6) £ 7 : 5⁴ + 3⁴ + 2⁴ = 3⁶ - 7

e*(3,4,6) = 0 : 31⁴ + 20⁴ - 17⁴ = 10⁶ and 89⁴ - 73⁴ - 24⁴ = 18⁶

e(3,5,6) £ 41 : 6⁵ + 6⁵ + 2⁵ = 5⁶ - 41

e*(3,5,6) £ 20 : 4⁵ - 3⁵ - 2⁵ = 3⁶ + 20

e(3,6,6) £ 219 : 5⁶ + 5⁶ + 5⁶ = 6⁶ + 219

e(3,7,6) £ 279 : 3⁷ + 3⁷ + 1⁷ = 4⁶ + 279, else 2⁷ + 2⁷ + 2⁷ = 3⁶ - 345

e*(3,7,6) £ 150 : 3⁷ + 3⁷ - 2⁷ = 4⁶ + 150

e(4,4,6) = 0 : 61⁴ + 56⁴ + 26⁴ + 4⁴ = 62⁴ + 47⁴ + 46⁴ + 8⁴ = 67⁴ + 44⁴ + 22⁴ + 8⁴ = 17⁶

e(4,5,6) £ 2 : 19⁵ + 12⁵ + 11⁵ + 10⁵ = 12⁶ - 2

e*(4,5,6) : no improvement

e(4,6,6) £ 430 : 12⁶ + 11⁶ + 11⁶ + 10⁶ = 14⁶ - 430

e(4,7,6) £ 217 : 2⁷ + 2⁷ + 2⁷ + 2⁷ = 3⁶ - 217

e*(4,7,6) £ 22 : 3⁷ + 3⁷ - 2⁷ - 2⁷ = 4⁶ + 22

e(5,4,6) = 0 : 16⁴ + 14⁴ + 10⁴ + 7⁴ + 6⁴ = 18⁴ + 10⁴ + 6⁴ + 6⁴ + 3⁴ = 18⁴ + 10⁴ + 7⁴ + 4⁴ + 2⁴ = 7⁶.

e(5,5,6) £ 1 : 5⁵ + 3⁵ + 3⁵ + 3⁵ + 3⁵ = 4⁶ + 1 and 19⁵ + 12⁵ + 11⁵ + 10⁵ + 1⁵ = 12⁶ - 1

e(5,6,6) £ 299 : 12⁶ + 11⁶ + 11⁶ + 10⁶ + 3⁶ = 14⁶ + 299

e(5,7,6) £ 89 : 2⁷ + 2⁷ + 2⁷ + 2⁷ + 2⁷ = 3⁶ - 89

e(6,5,6) = 0 : 19⁵ + 12⁵ + 11⁵ + 10⁵ + 1⁵ + 1⁵ = 12⁶ and 119⁵ + 81⁵ + 46⁵ + 41⁵ + 24⁵ + 14⁵ = 55⁶

e(6,6,6) £ 3 : 10⁶ + 9⁶ + 7⁶ + 7⁶ + 4⁶ + 3⁶ = 11⁶ + 3

e(6,7,6) £ 39 : 2⁷ + 2⁷ + 2⁷ + 2⁷ + 2⁷ + 2⁷ = 3⁶ - 39

e(7,6,6) = 0 : (1077⁶ + 894⁶ + 702⁶ + 474⁶ + 402⁶ + 234⁶ + 74⁶ = 1141⁶ (Lander & Parkin)

e(2,2,7) = 0 : 278² + 29² = 5⁷

e*(2,2,7) = 0 : 33² - 31² = 2⁷

e(2,3,7) £ 2 : 5³ + 1³ = 2⁷ - 2

e*(2,3,7) £ 1 : 7³ - 6³ = 2⁷ - 1

e(2,4,7) £ 14 : 23⁴ + 3⁴ = 16⁷ - 14

e*(2,4,7) : no improvement

e(2,5,7) £ 139 : 4⁵ + 4⁵ = 3⁷ - 139

e*(2,5,7) £ 83 : 3⁵ - 2⁵ = 2⁷ + 83

e(2,6,7) £ 30 : 5⁶ + 3⁶ = 4⁷ - 30

e(2,7,7) £ 1931 : 2⁷ + 2⁷ = 3⁷ - 1931, else 5⁷ + 5⁷ = 6⁷ - 123686

e(3,3,7) £ 1 : 5³ + 1³ + 1³ = 2⁷ - 1, 11³ + 8³ + 7³ = 3⁷ - 1 and 156³ + 80³ + 78³ = 9⁷ - 1

e*(3,3,7) = 0 : 7³ - 6³ + 1³ = 2⁷, 75³ - 52³ - 11³ = 6⁷ and 85³ - 77³ - 54³ = 2⁷

e(3,4,7) £ 2 : 23⁴ + 3⁴ + 2⁴ = 6⁷ + 2

e*(3,4,7) : no improvement

e(3,5,7) £ 7 : 21⁵ + 14⁵ + 11⁵ = 9⁷ + 7

e*(3,5,7) : no improvement

e(3,6,7) £ 29 : 5⁶ + 3⁶ + 1⁶ = 4⁷ - 29

e*(3,6,7) : no improvement

e(3,7,7) £ 1803 : 2⁷ + 2⁷ + 2⁷ = 3⁷ - 1803, else 3⁷ + 3⁷ + 3⁷ = 4⁷ - 9823

e(4,3,7) = 0 : 5³ + 1³ + 1³ + 1³ = 2⁷, 11³ + 8³ + 7³ + 1³ = 3⁷ and 56³ + 46³ + 19³ + 5³ = 6⁷

e(4,4,7) = 0 : 126⁴ + 111⁴ + 50⁴ + 22⁴ = 17⁷

e*(4,4,7) = 0 : 17⁴ - 14⁴ - 14⁴ - 9⁴ = 2⁷

e(4,5,7) £ 8 : 21⁵ + 14⁵ + 11⁵ + 1⁵ = 9⁷ + 8, else 4⁵ + 4⁵ + 2⁵ + 2⁵ = 3⁷ - 75

e*(4,5,7) £ 1 : 28⁵ + 26⁵ + 11⁵ - 25⁵ = 11⁷ - 1

e(4,6,7) £ 28 : 5⁶ + 3⁶ + 1⁶ + 1⁶ = 4⁷ - 28

e*(4,6,7) : no improvement

e(4,7,7) £ 1675 : 2⁷ + 2⁷ + 2⁷ + 2⁷= 3⁷ - 1675, else 3⁷ + 3⁷ + 3⁷ + 3⁷ = 4⁷ - 7636

e(5,4,7) £ 1 : 60⁴ + 45⁴ + 35⁴ + 31⁴ + 7⁴ = 11⁷ + 1

e(5,5,7) = 0 : 505⁵ + 315⁵ + 266⁵ + 212⁵ + 115⁵ = 87⁷

e(5,6,7) £ 13 : is 44⁶ + 38⁶ + 24⁶ + 10⁶ + 10⁶ = 27⁷ + 13

e(5,6,7) £ 1547 : 2⁷ + 2⁷ + 2⁷ + 2⁷ + 2⁷ = 3⁷ - 1547, else 9⁷ + 9⁷ + 6⁷ + 5⁷ + 5⁷ = 10⁷ + 2124

e(6,6,7) £ 4 : 24⁶ + 23⁶ + 19⁶ + 17⁶ + 5⁶ + 3⁶ = 17⁷ - 4

e(6,7,7) £ 1419 : 2⁷ + 2⁷ + 2⁷ + 2⁷ + 2⁷ + 2⁷ = 3⁷ - 1419

The best estimates for S(m,n) based on the above data are presented in the following table. An asterisk indicates that equality has not been proved.

Finally, the following table summarises S*(m,n).

URL : www.glasgowg43.freeserve.co.uk/chap45.htm