OEIS.org alerted me to the fact that 871 is the smallest number that is the sum of three triangular numbers 23 different ways. I set out to find those 23 ways. We’ll have to count 0 as the 0th triangular number for four of them, but here are those record-breaking 23 ways:

871 is the hypotenuse of Pythagorean triple 335-804-871 which is (5-12-13) times 67.

871 is also repdigit 111 in BASE 29 because 1(29²) + 1(29¹) + 1(29º) = 871.

• 871 is a composite number.
• Prime factorization: 871 = 13 × 67
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 871 has exactly 4 factors.
• Factors of 871: 1, 13, 67, 871
• Factor pairs: 871 = 1 × 871 or 13 × 67
• 871 has no square factors that allow its square root to be simplified. √871 ≈ 29.512709

Carnivals are part of most state fairs, including the Utah State Fair.

Here is a mathematically nonsensical commercial encouraging us to “Go Beyond Ordinary at the Utah State Fair – Sept 7 – 17, 2017.” (You will have to click this picture and the facebook post to see the video.)

I will be hosting the September 2017 Math Education Blog Carnival in a couple of weeks! I am excited, but also a bit terrified. I have never done anything like this before, and, fellow bloggers, I would really appreciate your support.

As always, the Math Education Blog Carnival will include posts about math that will make sense and be great fun for teachers, their students, and even parents. If you have a math education blog post, please submit it to this month’s blog carnival! Click here for instructions on how to submit your post. You can also contact me on twitter: Iva Sallay @findthefactors.

Now here’s a little bit about the number 866:

29² + 5² = 866, so 866 is the hypotenuse of a Pythagorean triple:

290-816-866 which is 2(29)(5) , 29² – 5² , 29² + 5² and 2 times (145-408-433)

• 866 is a composite number.
• Prime factorization: 866 = 2 × 433
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 866 has exactly 4 factors.
• Factors of 866: 1, 2, 433, 866
• Factor pairs: 866 = 1 × 866 or 2 × 433
• 866 has no square factors that allow its square root to be simplified. √866 ≈ 29.4278779

We’ve seen that the sum of the hypotenuse of a primitive Pythagorean triple and either of its legs is either a perfect square or two times a perfect square. What about just the legs themselves? Is there anything special about their sums? I think so. I think primitive Pythagorean triples have some terrific legs! Look at this chart and see if you agree.

If a sum is a prime number, there is only one way to get that sum. If the sum is repeated, then its prime factorization only uses prime numbers already on the list.

I’m not as impressed with the differences of the legs, but it is interesting that the same prime numbers show up:

863 is the sum of consecutive primes several ways:

• 863 is the sum of the fifteen prime numbers from 29 to 89.
• 863 is the sum of the thirteen prime numbers from 41 to 97.
• 107 + 109 + 113 + 127 + 131 + 137 + 139 = 863; that’s seven consecutive primes.
• 163 + 167+ 173 + 179 + 181 = 863; that’s five consecutive primes.

863(863 + 6) = 749,947, a palindrome. Thank you OEIS.org for that fun fact.

• 863 is a prime number.
• Prime factorization: 863 is prime.
• The exponent of prime number 863 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 863 has exactly 2 factors.
• Factors of 863: 1, 863
• Factor pairs: 863 = 1 × 863
• 863 has no square factors that allow its square root to be simplified. √863 ≈ 29.37686

How do we know that 863 is a prime number? If 863 were not a prime number, then it would be divisible by at least one prime number less than or equal to √863 ≈ 29.4. Since 863 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 863 is a prime number.

Mathematics is a wonderful collection of patterns.

If we look at the patterns of Pythagorean triples, can we predict how often 845 is the hypotenuse of a Pythagorean triple? Yes, we can.

The prime factorization of 845 is 5 × 13². Now 5, 13, 169 each has one primitive triple while 65 has two primitive triples. That’s a total of five primitive triples. Each of those primitives has a multiple with 845 as the hypotenuse.

1. 5’s primitive (3-4-5) times 169 is 507-676-845
2. 13’s primitive (5-12-13) times 65 is 325-780-845
3. 169’s primitive (119-120-169) times 5 is 595-600-845
4. 65’s first primitive (16-63-65) times 13 is 208-819-845
5. 65’s second primitive (33-56-65) times 13 is 429-728-845

Because ALL of its prime factors are primitive Pythagorean triple hypotenuses, 845 will also have some primitives of its own. How many?

Now 325, which is 5² × 13, has five non-primitive triples and two primitive triples, 36-323-325 and 204-253-325 . Since 325 and 845 have the same number of non-primitive triples, and similar looking prime factorizations, is it reasonable to assume that 845 will have two primitive triples as well? Yes, it is!

We can find 845’s primitive Pythagorean triples by looking at the ways to write 845 as the sum of two squares:

29² + 2² = 845, 26² + 13² = 845, and 22² + 19² = 845

26 is a multiple of 13 so it will duplicate one of the triples given by one of 845’s factors. The other two equations give us these two primitive Pythagorean triples:

1. 116-837-845, which was calculated from 2(29)(2), 29² – 2², 29² + 2²
2. 123-836-845, which was calculated from 22² – 19², 2(22)(19), 22² + 19²

Thus 845 has a total of 5 + 2 = 7 Pythagorean triples in which it is the hypotenuse!

What if we wanted to know how many times 5² × 13² = 4225 is the hypotenuse of a Pythagorean triple. Could we predict how many times that would be? Yes, we can!

Besides 1 and itself, the factors of 4225 are 5, 13, 25, 65, 169, 325, and 845, all Pythagorean triple hypotenuses. Each of their respective primitive Pythagorean triples has a multiple with 4225 as the hypotenuse:

1. 845 times 5’s primitive
2. 325 times 13’s primitive
3. 169 times 25’s primitive
4. 25 times 169’s primitive
5. 65 times 65’s two primitives
6. 13 times 325’s two primitives
7. 5 times 845’s two primitives

That’s a total of 10 non-primitive Pythagorean triples for 4225.

We know that 4225 will have some primitives of its own. If the pattern holds true, I hypothesize that 4225 will also have two primitives. So I ask, “What sums of two squares equal 4225?”

• 63² + 16² = 4225; 63 and 16 have no common prime factors so this will produce a primitive triple.
• 60² + 25² = 4225; 60 and 25 have 5 as a common factor so this will duplicate one of 4225’s other non-primitives.
• 56² + 33² = 4225; 56 and 33 have no common prime factors so this will produce a primitive triple.
• 52² + 39² = 4225; 52 and 39 have 13 as a common factor so this will duplicate one of 4225’s other non-primitives.

Thus the pattern held true and my hypothesis was correct! 4225 has a total of 10 + 2 = 12 Pythagorean triples! All of this has led me to some conclusions:

If x is a prime number with a remainder of 1 when divided by 4, and a is an integer that is greater than or equal to one, then xᴬ will have one primitive triple while the total number of Pythagorean triples will be A.

If x and y are prime numbers each with a remainder of 1 when divided by 4, and if A and B are integers with B ≥ A and A ≥ 1, then xᴬ × yᴮ will have two primitive triples. It appears that the total number of triples xᴬ × yᴮ will have will be A + B + 2Bᴬ. That is my conjecture.

Let’s check my conjecture for the numbers discussed in this post:

• 5¹ and 13¹ each have 1 triple
• 5² and 13² each have 2 triples
• 5² × 13¹ and 5¹ × 13² each have 1 + 2 + 2(2¹) = 7 triples
• 5² × 13² has 2 + 2 + 2(2²) = 12 triples

So far the conjecture appears to ring true. Perhaps I’ll explore it a little more when I write post 2 × 5² × 17¹ later this week.

Now here’s a little more about the number 845:

844, 845, 846, 847, and 848 are the smallest five consecutive numbers whose square roots can be simplified.

845 is palindrome 5A5 in BASE 12 (A is 10 base 10) because 5(12²) + 10(12¹) + 5(12º) = 845

845 is 500 in BASE 13 because 5(13²) = 845

• 845 is a composite number.
• Prime factorization: 845 = 5 × 13 × 13, which can be written 845 = 5 × 13²
• The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 × 3  = 6. Therefore 845 has exactly 6 factors.
• Factors of 845: 1, 5, 13, 65, 169, 845
• Factor pairs: 845 = 1 × 845, 5 × 169, or 13 × 65
• Taking the factor pair with the largest square number factor, we get √845 = (√169)(√5) = 13√5 ≈ 29.06888