Math 250 S2020, Number Theory

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Triangular Numbers Math 250 S2020, Number Theory

Worksheet No 2 (Includes WS No 1)

Name: Collaborators:

1. We introduced triangular numbers in class and found two formulas satisﬁed by triangular numbers. Using the dotted grid-paper, ﬁnd at least 5 more of such formulas.

2. Using the formula for triangular numbers

n(n + 1) t = (1) n 2 choose two of the formulas you discovered and give a convincing argument (a proof) that they are true for all possible value of n. The formula (1) can be taken as the deﬁnition of the triangular numbers. In the rest of the course, we will use this as its deﬁnition.

3. Do the following experiment: Add the first n odd integers ( first 1 odd integers, then the first 2 odd integers, then the first 3 odd integers). Go far enough until you have a clear conjecture (an educated guess) of what is the pattern 4. (Continuing with the previous problem) Using dots to form a geometric figure (similar to what we did in class with dots and triangle when we introduced the triangular numbers), give a convincing argument to show that your conjecture is true. 5. The number 1 and 36 are both perfect squares and triangular numbers. What is the next triangular number that is also a perfect square? (Hint: It’s a triangular number between 800 and 1200. Feel free to use a computer to help you do the calculations to find the next triangular number that is also a perfect square. you are not allowed to google (or be told) the answer 6. Using a list of triangular numbers find an expression that makes the following statements true • M is a triangular if and only if and only if is an odd perfect square. (Hint: the missing expression has the form aM + 1) • N is an odd perfect square, if and only if, is a triangular number.

7. Using formula (1) (the deﬁnition) prove that an integer n is a triangular number if and only if 8n + 1 is a perfect square. 8. Using the deﬁnition prove that the sum of any two consecutive triangular numbers is a perfect square. 9. Prove that if n is a triangular number, then so are 9n + 1, 25n + 3 and 49n + 6

2 10. The expression tn + tn−1 = n can be interpreted as saying: “The sum of two triangular numbers is a perfect square”. Find an interpretation for the formula

2 9(2n + 1) = t9n+4 − t3n+1

11. The difference between the squares of two consecutive triangular numbers is always a cube. looking at a list of triangular number and experimenting, find a precise formula expressing this fact. 12. In the sequence of triangular numbers, find the following: (a) Two triangular numbers whose sum and differences are also triangular numbers. (b) Three successive triangular numbers whose product is a perfect square. (c) Three successive triangular numbers whose sum is a perfect square.

Extra Problems

Instructor G. Polanco Triangular Numbers Math 250 S2020, Number Theory

A (Open ended) Above we found that 1, 36 and at least one more number are both triangular and perfect square: i.e these n’s are such that n(n + 1) = m2 2 Using this equality, can you develop some criteria for ﬁnding triangular numbers that are squares?

B Here is another criteria: If the triangular number tn is a perfect square, prove that t4n(n+1) is also a square. C Using the last excercise ﬁnd 3 examples of squares that are also triangular.

D Generalize problem 7, i.e. ﬁnd all a’s and b’s that make true the statement “n is triangular, if and only if an + b is a perfect square. E Generalize problem 9. i.e ﬁnd other expressions of the form an + b that would give a similar true statement with an + b in place of 25n + 3 (or the other expressions). Does a has to always be an odd perfect square ?

Instructor G. Polanco