Mathematics

426 My Response to a Pi-lish Question

/ ivasallay / 3 Comments
Hungarian Pi

A comma is used for decimals in many countries.

This last week there was a post on the Mathemagical Site titled “Do You Speak Pilish?”  Some people remember the digits of π by memorizing carefully constructed sentences in which the first word has three letters, the second word has one letter, and so on. Several examples were given, not just in English, but in eight other languages as well!

Hungarian was not one of the languages listed, but I wondered if there could possibly be a Pilish way for Hungarians to remember the digits of pi? (Almost all of my husband’s relatives were born in Hungary, and I am fascinated with the country and the language.) I just had to google “Magyar pi szám,” to find an article titled Minden idők legjobb magyar nyelvű pi-verse.

Now while I can read many Hungarian words, the sentence structure is so different from English that my comprehension isn’t as good as I’d like it to be. My son, David, taught himself the basics of the language before he went there to live and work several years ago. I emailed him the article requesting that he help me with the translation. In the email he sent back you will notice the problem with word for word translation of Hungarian into English. My son wrote:

“I don’t think I could translate it whilst maintaining the word lengths (which is the whole point). I’m giving it to you with a more or less word for word translation along with one that is written in more natural English. The Ludolph it mentions in the poem is the Dutch mathematician Ludolph van Ceulen, who was the first to publish pi up to 20 digits.”

I put his word for word translation in the following graphic:

Hungarian Pilish Pi

 

Here is David’s translation into more natural English:

  • Instead of the old and rough approximation,
  • Count the letters that come, word for word
  • If we end here at twenty words, we already have Ludolph’s result,
  • but exactly 10 more come from this last stanza.
  • That, I can promise confidently.”

Here is my answer to the question, “Do you speak Pilish?”

Not really. I am not the least bit interested in memorizing some cute paragraph in English to help me remember the first 30 or so digits of pi, BUT in Hungarian, I am going to give a try!

  • 426 is a composite number.
  • Prime factorization: 426 = 2 x 3 x 71
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 426 has exactly 8 factors.
  • Factors of 426: 1, 2, 3, 6, 71, 142, 213, 426
  • Factor pairs: 426 = 1 x 426, 2 x 213, 3 x 142, or 6 x 71
  • 426 has no square factors that allow its square root to be simplified. √426 ≈ 20.6398

423 and Level 4

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4 + 2 + 3 = 9 therefore 423 can be evenly divided by both 3 and 9. Thus we know right away that it is a composite number whose square root can be simplified.

Also ANY natural number ending in 24 or 25 has a reducible square root, so with 423, 424, and 425 we have THREE consecutive numbers with reducible square roots. Sadly, 426 breaks the pattern so we do not have a fourth.

423 Puzzle

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-09

  • 423 is a composite number.
  • Prime factorization: 423 = 3 x 3 x 47, which can be written 423 = (3^2) x 47
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 423 has exactly 6 factors.
  • Factors of 423: 1, 3, 9, 47, 141, 423
  • Factor pairs: 423 = 1 x 423, 3 x 141, or 9 x 47
  • Taking the factor pair with the largest square number factor, we get √423 = (√9)(√47) = 3√47 ≈ 20.5670

423 Logic

414 and Level 3

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414 is even and 4 + 1 + 4 = 9. These two facts help us to know that 414 can be evenly divided by 2, 3, 6, and 9. Its other factors are listed below the puzzle.

Today’s puzzle is really three puzzles in one. Solve some or all of them – your choice.

  • What are the missing factors?
  • Can you fill in the rest of the multiplication table?
  • What is the significance of the colors in the puzzle? Perhaps someone will type that answer in the comments because, otherwise, I’m not telling! (But you can look at puzzles 417 or 422 for more hints.)

414 Puzzle

Print the puzzles or type the factors on this black and white excel file: 10 Factors 2015-03-02

  • 414 is a composite number.
  • Prime factorization: 414 = 2 x 3 x 3 x 23, which can be written 414 = 2 x (3^2) x 23
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 x 3 x 2 = 12. Therefore 414 has exactly 12 factors.
  • Factors of 414: 1, 2, 3, 6, 9, 18, 23, 46, 69, 138, 207, 414
  • Factor pairs: 414 = 1 x 414, 2 x 207, 3 x 138, 6 x 69, 9 x 46, or 18 x 23
  • Taking the factor pair with the largest square number factor, we get √414 = (√9)(√46) = 3√46 ≈ 20.3470

 414 Factors

410 Is The Sum of Two Squares Two Different Ways

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410 = 59 + 61 + 67 + 71 + 73 + 79, six consecutive prime numbers.

410 = 199 + 211, two consecutive prime numbers.

  • 410 is a composite number.
  • Prime factorization: 410 = 2 x 5 x 41
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 410 has exactly 8 factors.
  • Factors of 410: 1, 2, 5, 10, 41, 82, 205, 410
  • Factor pairs: 410 = 1 x 410, 2 x 205, 5 x 82, or 10 x 41
  • 410 has no square factors that allow its square root to be simplified. √410 ≈ 20.2485

Those factors hardly seem like anything to write home about, but wait….

410 equals

How do I know all this? I applied something I read on Dr. David Mitchell’s blog, Lattice Labyrinths. He recently wrote a fascinating post about Tessellations and Pythagorean Triples. You have to read the article or at least look at his tessellations!

I didn’t know multiplying tessellating lattice labyrinths had anything to do with Pythagorean triples, but they do! (Actually I didn’t know anything about tessellating lattice labyrinths until I read the post.)

About the fourth or fifth paragraph of Dr. Mitchell’s post, he states something I didn’t know before: If you take one number that is the sum of two squares and find another number that is also the sum of two squares, and then if you multiply those two numbers together, that product will also be the sums of two squares! Since I knew both 10 and 41 could be written as the sum of two squares, I had to see if 410 could be also. I was doubly surprised and certainly not the least bit disappointed. I don’t know if it’s unusual that most of 410’s factors can also be written as the sum of two squares.

Because 410 and so many of its factors have that property, it is the hypotenuse of four non-primitive Pythagorean triples:

  • [266-312-410] is [133 – 156 – 205] times 2.
  • [246-328-410] is [3-4-5] times 82.
  • [168-374-410] is [84-187-205] times 2.
  • [90-400-410] is [9-40-41] times 10.

It is also the short leg of four other triples:

  • [410-984-1066] is [5-12-13] times 82.
  • [410-1656-1706] is [205-828-853] times 2.
  • [410-8400-8410] is [41-840-841] times 10.
  • [410-42024-42026] is [205-21012-21013] times 2.

 

408 and Level 6

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408 is divisible by 8 and so are 1408, 2408, 3408, 4408, . . . . . . and any other number whose last 3 digits are 408. This is true because if the last 3 digits of a number are divisible by 8, the entire number is also!

Also since 4 + 8 = 12, a multiple of 3, we know that 408 is divisible by 3.

408’s factors are listed below the puzzle.

408 Puzzle

Print the puzzles or type the factors on this excel file: 12 Factors 2015-02-23

  • 408 is a composite number.
  • Prime factorization: 408 = 2 x 2 x 2 x 3 x 17, which can be written 408 = (2^3) x 3 x 17
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 408 has exactly 16 factors.
  • Factors of 408: 1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 136, 204, 408
  • Factor pairs: 408 = 1 x 408, 2 x 204, 3 x 136, 4 x 102, 6 x 68, 8 x 51, 12 x 34, or 17 x 24
  • Taking the factor pair with the largest square number factor, we get √408 = (√4)(√102) = 2√102 ≈ 20.199

408 Logic

402 and Passing Along The Spectacular Blog Award

/ ivasallay / 4 Comments

We can find some of the factors of 402 rather quickly. It’s even, so 2 is a factor. 4 + 0 + 2 = 6, a multiple of 3, so 3 is also a factor. Six, then, would also be a factor. How many factors does 402 have and what are they? Scroll down to the end of the post to find out!

Nerd in the Brain created the-Spectacular-Blog-Award. This is how she describes it: “🙂It’s a simple award with no strings attached…no questions to answer, no specific number of people to nominate, no obligation. Recipients can just bask in the glory of knowing that another blogger thinks they’re super-awesome.” 🙂 

She continued, “This award is not just for me to give to people! If you want to let another blogger know that you think they’re fantastic, you go right ahead and snag this little award and pass it along to them. Share, share!” Keep Nerd in the Brain in the loop if you give someone this award.

So I for one am “snagging” this award to pass on to someone else:

Update: I gave the award to Abyssbrain whose blog, Mathemagicalsite.wordpress.com, has, unfortunately, been deleted.

The only thing I can think of to quell my disappointment is to find a couple more things about the number 402 and add them to this post:

Here are five ways 402 is the sum of three squares:

  • 20² + 1² + 1² = 402
  • 19² + 5² + 4² = 402
  • 17² + 8² + 7² = 402
  • 16² + 11² + 5² = 402
  • 13² + 13² + 8² = 402

402 is 123 in BASE 19 because 1(19²) + 2(19¹) + 3(19°) = 402.

  • 402 is a composite number.
  • Prime factorization: 402 = 2 x 3 x 67
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 402 has exactly 8 factors.
  • Factors of 402: 1, 2, 3, 6, 67, 134, 201, 402
  • Factor pairs: 402 = 1 x 402, 2 x 201, 3 x 134, or 6 x 67
  • 402 has no square factors that allow its square root to be simplified. √402 ≈ 20.0499

40% of Numbers Up To 400 Have Square Roots That Can Be Simplified

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  • 400 is a composite number.
  • Prime factorization: 400 = 2 x 2 x 2 x 2 x 5 x 5, which can be written 400 = (2^4) x (5^2)
  • The exponents in the prime factorization are 4 and 2. Adding one to each and multiplying we get (4 + 1)(2 + 1) = 5 x 3 = 15. Therefore 400 has exactly 15 factors.
  • Factors of 400: 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400
  • Factor pairs: 400 = 1 x 400, 2 x 200, 4 x 100, 5 x 80, 8 x 50, 10 x 40, 16 x 25, or 20 x 20
  • 400 is a perfect square. √400 = 20

A few months ago I made a chart showing the number of factors for the first 300 counting numbers. Since this is my 400th post, I’d like to include a chart showing the number of factors for all the numbers from 301 to 400. I’m also interested in consecutive numbers with the same number of factors and whether or not the square root of a number can be reduced. The red numbers have square roots that can be reduced.

301-400 Same Number of Factors

 

The longest streak of consecutive numbers with the same number of factors is only three. There are three sets of three consecutive numbers on this chart. (Between 200 and 300 there was a streak of four consecutive numbers with six factors each.)

How do the number of factors of these 100 numbers stack up against the previous 300? The following chart shows the number of integers with a specific number of factors and how many of those integers have reducible square roots:

1-400 Number of Factors

  • 39.5% or slightly less than 40% of the numbers up to 400 have reducible square roots.
  • Most of these numbers have 2, 4, or 8 factors. Numbers with two factors are prime numbers. Almost all numbers with four factors are the product of two different prime numbers, and nearly two-thirds of the numbers with eight factors are the product of three different prime numbers.
  • There isn’t much change between the percentages of reducible square roots from one list to the next.

396 What Pythagorean Triple Comes Next?

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396 is a multiple of 4, but not of 8, so just like 12, 20, 28, and 36, it is a leg of a primitive Pythagorean triple that is included in this infinite sequence of primitive triples (12–5–13), (20–21–29), (28–45–53), (36–77–85) . . . , which I’ve illustrated below:

20-21-29 What Pythagorean Triple Comes Next

Because they are Pythagorean triples, we know that 12² + 5² = 13², 20² + 21² = 29², 28² + 45² = 53², 36² + 77² = 85², and so forth.

What Pythagorean Triple Comes Next? Try to figure it out yourself, then scroll down a little bit to see if you are correct. In the meantime, let me tell you a little bit about the number 396:

  • 396 is a composite number.
  • Prime factorization: 396 = 2 x 2 x 3 x 3 x 11, which can be written 396 = (2^2) x (3^2) x 11
  • The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 x 3 x 2 = 18. Therefore 396 has exactly 18 factors.
  • Factors of 396: 1, 2, 3, 4, 6, 9, 11, 12, 18, 22, 33, 36, 44, 66, 99, 132, 198, 396
  • Factor pairs: 396 = 1 x 396, 2 x 198, 3 x 132, 4 x 99, 6 x 66, 9 x 44, 11 x 36, 12 x 33 or 18 x 22
  • Taking the factor pair with the largest square number factor, we get √396 = (√11)(√36) = 6√11 ≈ 19.8997

The next primitive Pythagorean triple in the sequence can be illustrated like this:

11-117-125

Let me tell you about five Pythagorean triples in which 396 is one of the legs:

  • The answer to which Pythagorean triple comes next was (44–117–125), and is illustrated above. If we multiply that triple by 9, we get (396-1053-1125).
  • Because 396 equals 36 x 11, another triple can be found by multiplying the previous primitive in the sequence (36–77–85) by 11 to get (396-847-935).
  • If we multiply the first triple in the sequence (12–5–13) by 33, we get (396-165-429).
  • The 16th primitive triple in the sequence is (132–1085–1093). If we multiply it by 3 we get (396-3255-3279).
  • The 49th Pythagorean triple in our sequence of primitive triples above has a short leg that could be illustrated with 396 yellow squares. That primitive Pythagorean triple is (396–9797–9805).

389 How to Know if a Prime Number is the Hypotenuse in a Primitive Pythagorean Triple

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  • 389 is a prime number.
  • Prime factorization: 389 is prime.
  • The exponent of prime number 389 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 389 has exactly 2 factors.
  • Factors of 389: 1, 389
  • Factor pairs: 389 = 1 x 389
  • 389 has no square factors that allow its square root to be simplified. √389 ≈ 19.723

How do we know that 389 is a prime number? If 389 were not a prime number, then it would be divisible by at least one prime number less than or equal to √389 ≈ 19.723. Since 389 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 389 is a prime number.

If a Pythagorean triple is a primitive, then its hypotenuse is always an odd number. Sometimes it is also a prime number. How can we know if a particular prime number, such as 389, is ever the hypotenuse of a Pythagorean triple?

Divide the last two digits of the prime number by 4 and look at the remainder:

  • If the remainder is 3, then that prime number will not ever be the hypotenuse of any Pythagorean triple.
  • If the remainder is 1, then it is the hypotenuse of a primitive Pythagorean triple! Find the proof here.

The last two digits of 389 is 89, and 89 ÷ 4 = 22 R1, so 389 and every other prime number ending in 89 is the hypotenuse of a primitive Pythagorean triple. What are the other two sides of this particular triangle? First we will figure out what two square numbers equal 389, then we will use those two numbers to figure out what two square numbers equal 389 squared:

Have you ever noticed that 1 is the first square number, 1 + 3 or 4 is also a square number, and 1 + 3 + 5 or 9 is yet another square number? This principle can be summarized by writing 1 + 3 + 5 + . . . + (2n – 1) = n squared.

Square Numbers - Sums of First Odd Numbers`

I’ll use this fact to determine when 389 is the hypotenuse of a primitive Pythagorean triple. I’ll subtract 1 from 389, then 3 from that result, and continue to subtract the next odd number until I get a perfect square. I should find the perfect square before I reach 194 (about half of 389).

So let’s begin:

389 to 289

At this point I know that 17² + another square equals 389. There are a few methods I can use to find the other perfect square.

  • I can find the other square by subtracting: 389 – 17²
  • I can take the last odd number used, 19, add one, then divide by two and then square that number.
  • Or I can continue subtracting odd numbers as before:

289 to 100

All of these methods give us 100 or 10² as the second perfect square. Notice that (33 + 1)/2 = 17 as well.

Now we have enough information to determine the other two numbers that make up the Pythagorean triple with 389 as the hypotenuse.

  • All of the numbers, a – b – c, in the triple can be found using the formulas Euclid provided more than 2000 years ago. Let 10 = n; and 17 = m.
  • a = m² – n² = 17² – 10² = 189 
  • b = 2mn = 2(10)(17) = 340
  • c = m² + n² = 17² + 10² = 389
  • thus 189 – 340 – 389 make a primitive Pythagorean triple. We know it is a primitive because 389 is prime and has no other factors besides one and itself.

385 Is a Square Pyramidal Number

/ ivasallay / 3 Comments

385 is the sum of the squares of the first ten counting numbers. Let me demonstrate what that means. All of the following are square pyramidal numbers:

  • 1² = 1
  • 1² + 2² = 5
  • 1² + 2² + 3² = 14
  • 1² + 2² + 3² + 4² = 30
  • 1² + 2² + 3² + 4² + 5² = 55
  • 1² + 2² + 3² + 4² + 5² + 6² = 91
  • 1² + 2² + 3² + 4² + 5² + 6² + 7² = 140
  • 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² = 204
  • 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² = 285
  • 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² = 385

We could have found the total, 385, much more quickly by putting the number ten in for “n” in this formula

Which means the sum of the first ten square counting numbers = (10 x 11 x 21)/6 = 5 x 11 x 7 = 385. (Notice its prime factorization!)

Here’s a video explanation:

Thanks to solvemymaths for bringing the video to my attention. It clearly explains why the formula works.

Here is the factoring information for 385:

  • 385 is a composite number.
  • Prime factorization: 385 = 5 x 7 x 11
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 385 has exactly 8 factors.
  • Factors of 385: 1, 5, 7, 11, 35, 55, 77, 385
  • Factor pairs: 385 = 1 x 385, 5 x 77, 7 x 55, or 11 x 35
  • 385 has no square factors that allow its square root to be simplified. √385 ≈ 19.621