Mathematics

1335 is a Pentagonal Number

/ ivasallay / Leave a comment

1335 is the 30th Pentagonal Number. Here’s a graphic to help you visualize that fact:

Here are some more facts about the number 1335:

  • 1335 is a composite number.
  • Prime factorization: 1335 = 3 × 5 × 89
  • 1335 has no exponents greater than 1 in its prime factorization, so √1335 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1335 has exactly 8 factors.
  • The factors of 1335 are outlined with their factor pairs in the graphic below.

1335 is also the hypotenuse of FOUR Pythagorean triples:
252-1311-1335 which is 3 times (84-437-445)
585-1200-1335 which is 15 times (39-80-89)
609-1188-1335 which is 3 times (203-396-445)
801-1068-1335 which is (3-4-5) times 267

What 1331 in Pascal’s Triangle Means

/ ivasallay / Leave a comment

1331 is the third row of Pascal’s triangle. What does that mean?
It means that 11³ = 1331,
and it means that 1 + 3 + 3 + 1 = 8 = 2³
Those third powers are not a coincidence.

It means that (x + y)³ = 1x³yº + 3x²y¹ + 3x¹y² + 1xºy³ or written more simply,
(x + y)³ = x³ + 3x²y¹ + 3x¹y² + y³

Likewise, (x – y)³ = 1x³yº – 3x²y¹ + 3x¹y² – 1xºy³ or simply
(x – y)³ = x³ – 3x²y¹ + 3x¹y² – y³

It also means that if you flip a coin three times, you’ll get
three heads and no tails 1 way: (HHH),
two head and one tail 3 ways: (HHT); (HTH); (THH),
one head and two tails 3 ways: (HTT); (THT); (TTH), and
no heads and three tails 1 way: (TTT).

That’s just some of what that third row being 1331 means but here are a few more facts about the number 1331.

  • 1331 is a composite number.
  • Prime factorization: 1331 = 11 × 11 × 11, which can be written 1331 = 11³
  • The exponent in the prime factorization is 3. Adding one, we get (3 + 1) = 4. Therefore 1331 has exactly 4 factors.
  • Factors of 1331: 1, 11, 121, 1331
  • Factor pairs: 1331 = 1 × 1331 or 11 × 121
  • Taking the factor pair with the largest square number factor, we get √1331 = (√121)(√11) = 11√11 ≈ 36.48287
  • 1331 is a perfect cube.

1331 is cool in some other bases, too:
It is 1000 in BASE 11, and
it’s 131 in BASE 35

1330 is the 19th tetrahedral Number

/ ivasallay / Leave a comment

The product of any three consecutive counting numbers is always divisible by 6. Why? Because one of the numbers has to be divisible by 3 and at least one number has to be divisible by 2. Dividing the product by 6 always results in a  tetrahedral number. 1330 is a good example:
(19)(20)(21)/6 = 1330

Since the first number in that product was 19, we know that 1330 is the 19th tetrahedral number, and it is the sum of the first 19 triangular numbers:

You can count all 1330 little green squares if you want in the graphic above if you choose.

Here are some more facts about the number 1330:

  • 1330 is a composite number.
  • Prime factorization: 1330 = 2 × 5 × 7 × 19
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1330 has exactly 16 factors.
  • Factors of 1330: 1, 2, 5, 7, 10, 14, 19, 35, 38, 70, 95, 133, 190, 266, 665, 1330
  • Factor pairs: 1330 = 1 × 1330, 2 × 665, 5 × 266, 7 × 190, 10 × 133, 14 × 95, 19 × 70, or 35 × 38
  • 1330 has no square factors that allow its square root to be simplified. √1330 ≈ 36.46917

1330 is the sum of the twenty-two prime numbers from 17 to 107. How cool is that?

1330 is the hypotenuse of a Pythagorean triple:
798-1064-1330 which is (3-4-5) times 266

1327 Record Breaking Prime Gap

/ ivasallay / Leave a comment

Here is a table of prime numbers with the gaps between them and the next prime number. When a prime gap is larger than any previous prime gap, I’ve highlighted it in yellow. As you can see 1327 breaks the previous record of 22 held by 1129.

That the next 33 numbers after 1327 are all composite is pretty remarkable. Even more remarkable is the fact that 1327 will hold the record for being the smallest prime number with the biggest prime gap until 9551 breaks it with a slightly larger prime gap of 36.

Here is some additional information about the number 1327:

  • 1327 is a prime number.
  • Prime factorization: 1327 is prime.
  • The exponent of prime number 1327 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1327 has exactly 2 factors.
  • Factors of 1327: 1, 1327
  • Factor pairs: 1327 = 1 × 1327
  • 1327 has no square factors that allow its square root to be simplified. √1327 ≈ 36.42801

How do we know that 1327 is a prime number? If 1327 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1327 ≈ 36.4. Since 1327 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1327 is a prime number.

The Shape of 1326

/ ivasallay / Leave a comment

1326 is the 51st triangular number because 26 × 51 = 1326 and 2(26) is 1 more than 51.

Since 51 is an odd number, 1326 is also a hexagonal number. It is the 26th hexagonal number because 26 is half of 52 which is 1 more than 51.

Triangular numbers aren’t hard to visualize, but hexagonal numbers might be. Here are 1326 tiny dots showing that 1326 is a hexagonal number:

Here are some more facts about the number 1326:

  • 1326 is a composite number.
  • Prime factorization: 1326 = 2 × 3 × 13 × 17
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1326 has exactly 16 factors.
  • Factors of 1326: 1, 2, 3, 6, 13, 17, 26, 34, 39, 51, 78, 102, 221, 442, 663, 1326
  • Factor pairs: 1326 = 1 × 1326, 2 × 663, 3 × 442, 6 × 221, 13 × 102, 17 × 78, 26 × 51, or 34 × 39
  • 1326 has no square factors that allow its square root to be simplified. √1326 ≈ 36.41428

Here are some factor trees for 1326 that won’t topple over.

1326 is also the long side of four different Pythagorean triple triangles:
840-1026-1326 which is 6 times (140-171-221)
126-1320-1326 which is 6 times (21-220-221)
510-1224-1326 which is (5-12-13) times 102
624-1170-1326 which is (8-15-17) times 78

1314 Desmos Art

/ ivasallay / Leave a comment

A teacher at my school had his students graph some polynomials and their inverses. I got to help some of his students with their graphs. After seeing the beautiful symmetry of the graphs together, I excitedly exclaimed to a few of the students, “Isn’t this a cool assignment?”

During my lunch, I put one of the graphs, its inverse, and some of their translations on Desmos and made a simple but lovely piece of art in the process. 

Before I was done, I showed it to a couple of students. One of them asked, “Are you saying that math can create art?” I loved replying, “Yes, it can!” Now that student wants to create some works of art, too. It was a privilege to show her how to use Desmos.

These are the inequalities I used to make my work of art:

MANY teachers have figured out that students could learn a lot about functions and their graphs by using Desmos to create drawings, pictures, or artwork. For example, look at this tweet and link shared by Chris Bolognese:

Now I’ll share some facts about the number 1314:

  • 1314is a composite number.
  • Primefactorization: 1314= 2 × 3 × 3 × 73,which can be written 1314 = 2 × 3² × 73
  • The exponents inthe prime factorization are 1, 2, and 1. Adding one to each and multiplying weget (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1314has exactly 12 factors.
  • Factors of 1314:1, 2, 3, 6, 9, 18, 73, 146, 219, 438, 657, 1314
  • Factor pairs: 1314= 1 × 1314,2 × 657, 3 × 438, 6 × 219, 9 × 146, or 18 × 73 
  • Taking the factorpair with the largest square number factor, we get √1314= (√9)(√146) = 3√146 ≈ 36.24914

1314 is the sum of two squares:
33² + 15² = 1314

1314 is the hypotenuse of a Pythagorean triple:
864-990-1314 which is 18 times (48-55-73) and
can also be calculated from 33² – 15², 2(33)(15), 33² + 15²

1306 is a Centered Triangular Number

/ ivasallay / Leave a comment

1306 tiny squares can be arranged in a triangle as I have done in the graphic below. This arrangement makes 1306 a centered triangular number:

Here’s more information about the number 1306:

  • 1306 is a composite number.
  • Prime factorization: 1306 = 2 × 653
  • 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 1306 has exactly 4 factors.
  • Factors of 1306: 1, 2, 653, 1306
  • Factor pairs: 1306 = 1 × 1306 or 2 × 653
  • 1306 has no square factors that allow its square root to be simplified. √1306 ≈ 36.13862

OEIS.org informs us that 1306 = 1¹ + 3² + 0³ + 6⁴.

1306 is the sum of two squares:
35² + 9² = 1306

1306 is the hypotenuse of a Pythagorean triple:
630-1144-1306 calculated from 2(35)(9), 35² – 9², 35² + 9²

1301 Cool Patterns of Centered Square Numbers

/ ivasallay / Leave a comment

1301 is a centered square number because it is the sum of two consecutive square numbers. It also happens to be exactly 100 more than the previous centered square number.

1301 is special because it is the sum of two consecutive squares, but it is not the only sum of two squares that is 100 more than another sum of two squares. Here is a chart of some of those squares with some other information. What patterns do you see?

What role did the triangular numbers play on that chart? The pattern you see of 4, 12, 24, etc. is not new. Look at this chart of centered square numbers below. Below each centered square number (in blue) are related sums of squares. The difference of those related sums and the centered square above it follows that same 4, 12, 24, . . .  pattern.

When you look just at centered square numbers, there is a different pattern for them as well. For example, if you know that 1301 is the 26th centered square number, then you can be sure that 1301 + 4(26) = 1405 will be the next centered square number.

When the chart above is “bent in half” so the list of centered square numbers begins at the lower left and ends at the lower right, there is yet another pattern. Look at the differences horizontally to see a cool pattern:

I’ve made it clear that centered square numbers are made when consecutive square numbers are added, such as 26² + 25² = 1301, but what does one look like? Here’s a picture of 1301 tiny little squares arranged as a centered square.

There are 100 tiny green squares around it because it is 100 more than the previous centered square number.

Here are some more facts about the number 1301:

  • 1301 is a prime number and the first prime in a prime triplet.
  • Prime factorization: 1301 is prime.
  • The exponent of prime number 1301 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1301 has exactly 2 factors.
  • Factors of 1301: 1, 1301
  • Factor pairs: 1301 = 1 × 1301
  • 1301 has no square factors that allow its square root to be simplified. √1301 ≈ 36.06938

How do we know that 1301 is a prime number? If 1301 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1301 ≈ 36.1. Since 1301 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1301 is a prime number.

Every centered square number is a hypotenuse of a Pythagorean triple. For example,
primitive 51-1300-1301 is calculated from 26² – 25², 2(26)(25), 26² + 25²

Here’s another way we know that 1301 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 26² + 25² = 1301 with 26 and 25 having no common prime factors, 1301 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1301 ≈ 36.1. Since 1301 is not divisible by 5, 13, 17, or 29, we know that 1301 is a prime number.

1296 Last Digit Effort

/ ivasallay / Leave a comment

If you always use a calculator to find the roots of perfect powers like 1296, then you will likely miss out on discovering the wonderful patterns those powers have. Look at the table below. What do you notice? What do you wonder?

When I looked at it, I noticed several patterns and I wondered about a few things that I explored to see if they were true. However, I won’t tell you what I noticed or what conjectures I made until the end of this post, because I don’t want to spoil YOUR chance to discover those patterns yourself. I will tell you that there are MANY patterns in the table above and that most of my conjectures were true, but one of them turned out not to be.

Besides the patterns that you will have to discover for yourself, the table can be very useful: You can use the table to find several fractional powers of the numbers in the body of the table. For example, if you wanted to find out what is 1296^(7/4), then you could simply run your finger down the n⁷ column and the n⁴ column simultaneously until you reach 1296 on the n⁴ column (because 4 is the denominator of the fractional power). Your answer, 279936, will be in the n⁷ column on the same row as 1296.

However, if you wanted to find 1296^(3/2), you would either have to expand the table to include 36² and 36³, or you would have to use the fact that 3/2 = 6/4. Then you could use the process of the previous paragraph to see that
1296^(3/2) = 1296^(6/4) = 46656.

Now I’ll share some more facts about the number 1296:

  • 1296 is a composite number, a perfect square, and a perfect fourth power.
  • Prime factorization: 1296 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3, which can be written 1296 = 2⁴ × 3⁴
  • The exponents in the prime factorization are 4 and 4. Adding one to each and multiplying we get (4 + 1)(4 + 1) = 5 × 5 = 25. Therefore 1296 has exactly 25 factors.
  • Factors of 1296: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 81, 108, 144, 162, 216, 324, 432, 648, 1296
  • Factor pairs: 1296 = 1 × 1296, 2 × 648, 3 × 432, 4 × 324, 6 × 216, 8 × 162, 9 × 144, 12 × 108, 16 × 81, 18 × 72, 24 × 54, 27 × 48, or 36 × 36
  • Taking the factor pair with the largest square number factor, we get √1296 = (√36)(√36) = 36.

1296 is the smallest number with exactly 25 factors.

There are MANY different factor trees you could make for 1296, but here are some made with squares:

1296 is the 21st Friedman number because of its digits and 9²×16 = 1296. (The same digits are on both sides of that equation.)

Since 36 is a triangular number and 36² = 1296, we get this wonderful, powerful fact:
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)² = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ = 1296

1296 looks interesting and often rather square when it is written in several bases:
It’s1210000 in BASE 3,
10000 in BASE 6,
2420 in BASE 8,
1700 in BASE 9,
1296 in BASE 10
900 in BASE 12,
789 in BASE 13 (Ha, ha. Seven ate nine!),
484 in BASE 17,
400 in BASE 18,
169 in BASE 33,
144 in BASE 34,
121 in BASE 35, and
100 in BASE 36

Do you know the square roots in base 10 of all the numbers in bold? Indeed, 1296 is a fascinating square number!

Here are some things I noticed on the table of powers for numbers ending in . . .

  • 1: The last digit is always 1
  • 2: The pattern for the last digit is 2, 4, 8, 6, repeating
  • 3: The pattern for the last digit is 3, 9, 7, 1, repeating
  • 4: The pattern for the last digit is 4, 6, 4, 6, repeating
  • 5: The last digit is always 5
  • 6: The last digit is always 6
  • 7: The pattern for the last digit is 7, 9, 3, 1, repeating (3’s pattern backward but starting with 7)
  • 8: The pattern for the last digit is 8, 4, 2, 6, repeating (2’s pattern backward but starting with 8)
  • 9: The pattern for the last digit is 9, 1, 9, 1, repeating
  • 0: The last digit is always 0

For the squares:

  • 1² and 9² both end in 1, and 1 + 9 = 10
  • 2² and 8² both end in 4, and 2 + 8 = 10
  • 3² and 7² both end in 9, and 3 + 7 = 10
  • 4² and 6² both end in 6, and 4 + 6 = 10

For the cubes:

  • 1³ ends in 1, and 9³ ends in 9
  • 2³ ends in 8, and 8³ ends in 2; 2 + 8 = 10
  • 3³ ends in 7, and 7³ ends in 3; 3 + 7 = 10
  • 4³ ends in 4, and 6³ ends in 6

Other observations:

  • No matter what n is, the last digit of n, n⁵, and n⁹ are always the same. Yes, that pattern continues with n¹³ and so forth.
  • The number of digits for 10ᵃ is a+1, and the number of digits for 9ᵃ is a.

Dan Bach noticed something else.

I had never heard of Benford’s Law before, so I’ve learned something new here!  All of us can learn something new by observing patterns and listening to each other.