Natural Numbers as the Sum of Natural Numbers – Part 1

No more impressive warning can be given to those who would confine knowledge and research to what is apparently useful, than the reflection that conic sections were studied for eighteen hundred years merely as an abstract science, without regard to any utility other than to satisfy the craving for knowledge on the part of mathematicians, and that then at the end of this long period of abstract study, they were found to be the necessary with which to attain the knowledge of the most important laws of nature.

Alfred North WHITEHEAD

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Generating natural numbers from arithmetical progressions and representing them with geometrical forms, originated and shaped invariably early number theory. The formation of triangular numbers 1, 1 + 2, 1 + 2 + 3, . . . and of square numbers 1, 1 + 3, 1 + 3 + 5, . . . are the most well-known examples.

Consecutive natural numbers are natural numbers n₁ and n₂ such that n₁ ∼ n₂ = 1. The aim of this blogpost is to explore the possibilities of representing first fifty natural numbers in all possible arithmetic series (with maximum common difference 2) as

(i) sums of consecutive natural numbers;

(ii) sums of consecutive odd natural numbers;

(iii) sums of consecutive even natural numbers.

To do this, we have used the following results.

Result 1.No number of the form 2^{(n – 1)}, where n is a positive integer, is expressible as a sum of consecutive natural numbers.

Result 2. Every odd positive number can be written as the sum of two consecutive natural numbers.

Result 3. Every squared number n = d² can be written as the sum of the first d odd natural numbers:

N = d² = 1 + 3 + 5 + ∙∙∙ + (2d – 1).

Result 4. No prime is expressible as a sum of successive odd positive numbers or as a sum of successive even positive numbers.

Result 5. Every natural number N with at least one odd divisor d > 1, is expressible as a sum of consecutive integers.

Result 6. Every natural number of the form n(n + 1) /2 can be expressed as the sum of the first n positive integers.

It is not possible to express 2 = 2¹ as the sum of consecutive natural numbers.

3 = 1 + 2

It is not possible to express 4 = 2² as the sum of consecutive natural numbers. But

4 = 1 + 3

5 = 2 + 3

6 = 1 + 2 + 3

6 = 2 + 4

7 = 3 + 4

It is not possible to express 8 = 2³ as the sum of consecutive natural numbers. But

8 = 3 + 5

9 = 4 + 5

9 = 1 + 3 + 5

10 = 1 + 2 + 3 + 4

10 = 4 + 6

11 = 5 + 6

12 = 3 + 4 + 5  

12 = 5 + 7

12 = 2 + 4 + 6

13 = 6 + 7

14 = 2 + 3 + 4 + 5

14 = 6 + 8

15 = 7 + 8

15 = 4 + 5 + 6

15 = 1 + 2 + 3 + 4 + 5

15 = 3 + 5 + 7

It is not possible to express 16 = 2⁴ as the sum of consecutive natural numbers. But

16 = 7 + 9

16 = 1 + 3 + 5 + 7

17 = 8 + 9

18 = 5 + 6 + 7

18 = 3 + 4 + 5 + 6

18 = 8 + 10

18 = 4 + 6 + 8

19 = 9 + 10

20 = 2 + 3 + 4 + 5 + 6

20 = 9 + 11

20 = 2 + 4 + 6 + 8

21 = 10 + 11

21 = 6 + 7 + 8     

21 = 1 + 2 + 3 + 4 + 5 + 6

21 = 5 + 7 + 9

22 = 4 + 5 + 6 + 7

22 = 10 + 12

23 = 11 + 12

24 = 7 + 8 + 9

24 = 11 + 13

24 = 6 + 8 + 10

24 = 3 + 5 + 7 + 9

25 = 12 + 13

25 = 3 + 4 + 5 + 6 + 7

25 = 1 + 3 + 5 + 7 + 9

26 = 5 + 6 + 7 + 8

26 = 12 + 14

27 = 13 + 14       

27 = 8 + 9 + 10  

27 = 2 + 3 + 4 + 5 + 6 + 7

27 = 7 + 9 + 11

28 = 1 + 2 + 3 + 4 + 5 + 6 + 7

28 = 13 + 15

28 = 4 + 6 + 8 + 10

29 = 14 + 15

30 = 9 + 10 + 11

30 = 6 + 7 + 8 + 9

30 = 4 + 5 + 6 + 7 + 8

30 = 14 + 16

30 = 8 + 10 + 12

30 = 2 + 4 + 6 + 8 + 10

31 = 15 + 16

It is not possible to express 32 = 2⁵ as the sum of consecutive natural numbers. But

32 = 15 + 17

32 = 5 + 7 + 9 + 11

33 = 16 + 17

33 = 10 + 11 + 12

33 = 3 + 4 + 5 + 6 + 7 + 8

33 = 9 + 11 + 13

34 = 7 + 8 + 9 + 10

34 = 16 + 18

35 = 17 + 18

35 = 5 + 6 + 7 + 8 + 9

35 = 2 + 3 + 4 + 5 + 6 + 7 + 8

35 = 3 + 5 + 7 + 9 + 11

36 = 11 + 12 + 13

36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8

36 = 17 + 19

36 = 10 + 12 + 14

36 = 6 + 8 + 10 + 12

36 = 1 + 3 + 5 + 7 + 9 + 11

37 = 18 + 19

38 = 8 + 9 + 10 + 11

38 = 18 + 20

39 = 19 + 20

39 = 12 + 13 + 14

39 = 4 + 5 + 6 + 7 + 8 + 9

39 = 11 + 13 + 15

40 = 6 + 7 + 8 + 9 + 10

40 = 19 + 21

40 = 7 + 9 + 11 + 13

40 = 4 + 6 + 8 + 10 + 12

41 = 20 + 21

42 = 13 + 14 + 15

42 = 9 + 10 + 11 + 12

42 = 3 + 4 + 5 + 6 + 7 + 8 + 9

42 = 20 + 22

42 = 12 + 14 + 16

42 = 2 + 4 + 6 + 8 + 10 + 12

43 = 21 + 22

44 = 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9

44 = 21 + 23

44 = 8 + 10 + 12 + 14

45 = 22 + 23

45 = 14 + 15 + 16

45 = 7 + 8 + 9 + 10 + 11

45 = 5 + 6 + 7 + 8 + 9 + 10

45 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9

45 = 13 + 15 + 17

45 = 5 + 7 + 9 + 11 + 13

46 = 10 + 11 + 12 + 13

46 = 22 + 24

47 = 23 + 24

48 = 15 + 16 + 17

48 = 23 + 25

48 = 14 + 16 + 18

48 = 9 + 11 + 13 + 15

48 = 3 + 5 + 7 + 9 + 11 + 13

49 = 24 + 25

49 = 4 + 5 + 6 + 7 + 8 + 9 + 10

49 = 1 + 3 + 5 + 7 + 9 + 11 + 13

50 = 11 + 12 + 13 + 14

50 = 8 + 9 + 10 + 11 + 12

50 = 24 + 26

50 = 6 + 8 + 10 + 12 + 14

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