Yunzhong sent from Aofei Temple
Quantum Bit Report | Public Account QbitAI

Now, 42 has also been breached.

It can be written as the sum of the cubes of 3 integers, which is the last "easter egg" of natural numbers within 100.

The honour belongs to Andrew Sutherland of MIT and Andrew Booker of the University of Bristol. Yes, two Andrews with the same name worked together on this mathematical breakthrough.

And published the results on the MIT Mathematics website:

It also attracted the attention of mathematics experts and enthusiasts. Fields Medal winner and Cambridge University professor Timothy Gowers also retweeted his congratulations on this achievement.

But what’s the point of this?

Of course, it is not the "Ultimate Mystery of the Universe: 42" in the science fiction movie "The Hitchhiker's Guide to the Galaxy".

Sum of 3 cubes

The problem first began in 1992, when mathematician Roger Heath-Brown conjectured that all natural numbers could be written as the sum of the cubes of three numbers.

But as time goes by, the rules are constantly deduced:

Except for natural numbers of the type 9n±4, all natural numbers within 100 can be written as the sum of the cubes of three integers.

But until 2015, there were still three natural numbers within 100: 33, 42 and 74, which remained unresolved and had no conclusion.

Later, mathematician Sander Huisman found 74 :

And in March of this year, 33 also had the answer:

Mathematician Tim Browning updated the result on his personal homepage.

By now, 42 has also been cracked, and it is said that the idea also benefited from the solution of 33.

And the more direct result is that the last such number within 100 has an answer. There is a theorem about the expression that natural numbers are the sum of the cubes of 3 integers:

Except for natural numbers of the type 9n±4, all natural numbers within 100 can be written as the sum of the cubes of three integers.

When the range is expanded to within 1000, there are still 10 natural numbers.

114, 165, 390, 579, 627, 633, 732, 906, 921 and 975.

The sum of 3 cubes within 100

As for how natural numbers within 100 are specifically expressed as the sum of 3 cubes, the public account MathDuoDaa has listed them, which we will copy as follows:

(Note: non-zero solution, choose one of multiple ways to write it)

1 = (-1)³ + 1³ + 1³

2 = 7³ + (-5)³ + (-6)³

3 = 1³ + 1³ + 1³

4 Impossible

5 Impossible

6 = (-1)³ + (-1)³ + 2³

7 = 104³ + 32³ + (-105)³

8 = (-1)³ + 1³ + 2³

9 = 217³ + (-52)³ + (-216)³

10 = 1³ + 1³ + 2³

11 = (-2)³ + (-2)³ + 3³

12 = 7³ + 10³ + (-11)³

13 Impossible

14 Impossible

15 = (-1)³ + 2³ + 2³

16 = (-511)³ + (-1609)³ + 1626³

17 = 1³ + 2³ + 2³

18 = (-1)³ + (-2)³ + 3³

19 = 19³ + (-14)³ + (-16)³

20 = 1³ + (-2)³ + 3³

21 = (-11)³ + (-14)³ + 16³

22 Impossible

23 Impossible

24 = (-2901096694)³ + (-1555055555)³ + 15584139827³

25 = (-1)³ + (-1)³ + 3³

26 = 297³ + 161³ + (-312)³

27 = (-1)³ + 1³ + 3³

28 = 14³ + 13³ + (-17)³

29 = 1³ + 1³ + 3³

30 = (-283059965)³ + (-2218888517)³ + 2220422932³

31 Impossible

32 Impossible

33 = 8866128975287528³ + (-8778405442862239)³ + (-2736111468807040)³

34 = (-1)³ + 2³ + 3³

35 = 14³ + (-8)³ + (-13)³

36 = 1³ + 2³ + 3³

37 = 50³ + 37³ + (-56)³

38 = 1³ + (-3)³ + 4³

39 = 117367³ + 134476³ + (-159380)³

40 Impossible

41 Impossible

42 = (-80538738812075974)³ + 80435758145817515³ + 12602123297335631³

43 = 2³ + 2³ + 3³

44 = (-5)³ + (-7)³ + 8³

45 = 2³ + (-3)³ + 4³

46 = (-2)³ + 3³ + 3³

47 = 6³ + 7³ + (-8)³

48 = (-23)³ + (-26)³ + 31³

49 Impossible

50 Impossible

51 = 602³ + 659³ + (-796)³

52 = 23961292454³ + 60702901317³ + (-61922712865)³

53 = (-1)³ + 3³ + 3³

54 = (-7)³ + (-11)³ + 12³

55 = 1³ + 3³ + 3³

56 = (-11)³ + (-21)³ + 22³

57 = 1³ + (-2)³ + 4³

58 Impossible

59 Impossible

60 = (-1)³ + (-4)³ + 5³

61 = 845³ + 668³ + (-966)³

62 = 3³ + 3³ + 2³

63 = 7³ + (-4)³ + (-6)³

64 = (-1)³ + 1³ + 4³

65 = 91³ + 85³ + (-111)³

66 = 1³ + 1³ + 4³

67 Impossible

68 Impossible

69 = 2³ + (-4)³ + 5³

70 = 11³ + 20³ + (-21)³

71 = (-1)³ + 2³ + 4³

72 = 7³ + 9³ + (-10)³

73 = 1³ + 2³ + 4³

74 = (-284650292555885)³ + (66229832190556)³ + (283450105697727)³

75 = 4381159³ + 435203083³ + (-435203231)³

76 Impossible

77 Impossible

78 = 26³ + 53³ + (-55)³

79 = (-19)³ + (-33)³ + 35³

80 = 69241³ + 103532³ + (-112969)³

81 = 10³ + 17³ + (-18)³

82 = (-11)³ + (-11)³ + 14³

83 = (-2)³ + 3³ + 4³

84 = (-8241191)³ + (-41531726)³ + 41639611³

85 Impossible

86 Impossible

87 = (-1972)³ + (-4126)³ + 4271³

88 = 3³ + (-4)³ + 5³

89 = 6³ + 6³ + (-7)³

90 = (-1)³ + 3³ + 4³

91 = 364³ + 192³ + (-381)³

92 = 1³ + 3³ + 4³

93 = (-5)³ + (-5)³ + 7³

94 Impossible

95 Impossible

96 = 10853³ + 13139³ + (-15250)³

97 = (-1)³ + (-3)³ + 5³

98 = 14³ + 9³ + (-15)³

99 = 2³ + 3³ + 4³

100 = 7³ + (-3)³ + (-6)³

significance

Perhaps after seeing this, you will also ask what is the significance of this result and rule?

Sorry, there is no theorem-like answer yet.

However, today's result of 42 has already excited a number of mathematicians and enthusiasts.

Perhaps the joy of discovery is also a kind of meaning.

What do you think?

Relevant information:

http://math.mit.edu/~drew/

https://aperiodical.com/2019/09/42-is-the-answer-to-the-question-what-is-80538738812075974%c2%b3-80435758145817515%c2%b3-12602123297335631%c2%b3/

For the first time, humans wrote 33 as the sum of the cubes of three integers:

https://mp.weixin.qq.com/s/7_lO1bIZynWX8_NNZviV5g