Which number can be expressed as the sum of two triangular numbers? I don't even know how to start with this one. Here is some data:(adsbygoogle = window.adsbygoogle || []).push({});

If you don't count 0 as a triangular number, then the following can:

2

4

6

7

9

11

12

13

16

18

20

21

22

24

25

27

29

30

31

34

36

37

And these can't:

3

5

8

10

14

15

17

19

23

26

28

32

33

35

If you do count 0 as one, then the following can:

2

3

4

6

7

9

10

11

12

13

15

16

18

20

21

22

24

25

27

28

29

30

31

34

36

37

And these can't:

5

8

14

17

19

23

26

32

33

35

Unlike the sum of two squares problem, you don't get anything so nice like if a and b are the sum of two triangles, then so is ab. Also, with sums of two squares, there's the fact if p is an odd prime, then p is a sum of two squares iff p = 1 (mod 4). Again, nothing as nice appears to be true for triangle numbers (even if you replace mod 4 with mod 3 or other small primes, or so it seems).

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# Homework Help: Sum of Two Triangular Numbers

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