Mathematics

1277 Strată Bolyai János in Timișoara, Romania

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Around the turn of the 20th century, Bolyai Farkás taught mathematics at a university in Transylvania.  One day he was too sick to teach, so he sent his mathematically gifted 13-year-old son, János, to teach his classes! As you might imagine, János became quite the mathematician in his own right.

Ninety-five years ago today Bolyai János went to Timișoara, Romania to announce his findings concerning geometry’s fifth postulate. For centuries it was argued that this parallel lines postulate could probably be proved using the previous four of Euclid’s postulates, and it should, therefore, be considered a theorem rather than a postulate. Bolyai János proved that it is indeed something that must be assumed rather than proven, because, by assuming it wasn’t necessary, he was able to create a new and very much non-Euclidean geometry, now known as hyperbolic geometry or Bolyai–Lobachevskian geometry.

Last summer I was walking with some family members through a shopping area behind the opera house in Timișoara, Romania. Suddenly my son, David, excitedly shouted, “Mom, look!” There we stood in front of a street sign marking the strată named for Bolyai János! Here is a picture of me in front of that street sign.

Under his image are several plaques. The first is a replica of part of his proof. Underneath are plaques with a quote from him translated into several languages. Perhaps your favorite language is among them. Here is a close-up of the plaques:

The plaque at the bottom is in English, “From nothing I have created a new and another world. It was with these words that on November 3, 1823, Janos Bolyai announced from Timișoara the discovery of the fundamental formula of the first non-Euclidean geometry.”

We did not get to visit the university named for Bolyai János, but I am thrilled that my son spotted this historic location!

Now I’ll write a little about the number 1277:

  • 1277 is a prime number.
  • Prime factorization: 1277 is prime.
  • The exponent of prime number 1277 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1277 has exactly 2 factors.
  • Factors of 1277: 1, 1277
  • Factor pairs: 1277 = 1 × 1277
  • 1277 has no square factors that allow its square root to be simplified. √1277 ≈ 35.73514

How do we know that 1277 is a prime number? If 1277 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1277 ≈ 35.7. Since 1277 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1277 is a prime number.

1277 is the sum of two squares:
34² + 11² = 1277

1277 is the hypotenuse of a Pythagorean triple:
748-1035-1277 calculated from 2(34)(11), 34² – 11², 34² + 11²

Here’s another way we know that 1277 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 34² + 11² = 1277 with 34 and 11 having no common prime factors, 1277 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1277 ≈ 35.7. Since 1277 is not divisible by 5, 13, 17, or 29, we know that 1277 is a prime number.

1276 is the Third Number in a Row with Exactly 12 Factors

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1276 is the 125th number to have exactly 12 factors. I’ve made a list of those numbers in the graphic below. Two consecutive numbers appearing on the list have only happened three times before. Those numbers are highlighted in blue. 1276 is special because with it, for the first time THREE consecutive numbers appear on the list!

Look at the prime factorizations of those three consecutive numbers:
1274 = 2·7²·13
1275 = 3·5²·17
1276 = 2²·11·29

How are they the same? Can you figure out a reason why they all have exactly 12 factors?

By the way, prime number 1277 spoiled the streak especially since 1278 = 2·3²·71 and also has 12 factors!

If you came up with a rule, I think you should know that not all numbers with 12 factors will follow that same rule. For example,
2³·3² = 72 and has 12 factors because 4·3=12.
2⁵·3 = 96 and has 12 factors because 6·2 = 12.

I hope that strengthens your hypothesis instead of destroying it!

Now I’ll tell you some more facts about the number 1276:

  • 1276 is a composite number.
  • Prime factorization: 1276 = 2 × 2 × 11 × 29, which can be written 1276 = 2² × 11 × 29
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1276 has exactly 12 factors.
  • Factors of 1276: 1, 2, 4, 11, 22, 29, 44, 58, 116, 319, 638, 1276
  • Factor pairs: 1276 = 1 × 1276, 2 × 638, 4 × 319, 11 × 116 22 × 58, or 29 × 44
  • Taking the factor pair with the largest square number factor, we get √1276 = (√4)(√319) = 2√319 ≈ 35.72114

1276 is the sum of 12 consecutive prime numbers:
79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 = 1276

1276 is the hypotenuse of a Pythagorean triple:
880-924-1276 which is (20-21-29) times 44

Finally, from OEIS.org, we learn that 1276 = 1111 + 22 + 77 + 66.

1275 is the 50th Triangular Number AND the 500th Number Whose Square Root Can Be Simplified

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I found two reasons to celebrate the number 1275: It is the 50th Triangular number, and it is the 500th number whose square root can be simplified.

First I’ll celebrate its square root by listing the 401st to the 500th numbers with simplifiable square roots. Having three or more simplifiable square roots in a row doesn’t happen that often, so I like to highlight them when it happens. 1274 and 1275 are highlighted because 1276 also has a square root that can be simplified:

If you’re wondering what are the first 400 numbers with simplifiable square roots, you can click on the graphics below that will give you 100 at a time:

1st 100 reducible square roots 2nd 100 reducible square roots Reducible Square Roots 516-765

Now to celebrate that 1275 is the 50th triangular number, I’ve arranged $12.75 in a triangle:

1275 can also be evenly divided by 5, and 25, in other words, by nickels and quarters!

Nickels won’t make a triangle but they can form a trapezoid. Here’s how I made this one: 1275 ÷ 5 = 255 which is 300 (the 24th triangular number) minus 45 (the 9th triangular number). Thus we can make $12.75 by arranging 255 nickels in a trapezoid with a top base of 10, a bottom base of 24, and a height of 15.

We can also use quarters to make a trapezoid. Here’s what I did: 1275 ÷ 25 = 51 which is 66 (the 11th triangular number) minus 15 (the 5th triangular number). Thus, $12.75 can be made by arranging 51 quarters in a trapezoid with a top base of 6, a bottom base of 11, and a height of 6.

Can you find any rectangular ways to arrange the coins to total $12.75?

Here’s a little more about the number 1275:

  • 1275 is a composite number.
  • Prime factorization: 1275 = 3 × 5 × 5 × 17, which can be written 1275 = 3 × 5² × 17
  • 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 × 3 × 2 = 12. Therefore 1275 has exactly 12 factors.
  • Factors of 1275: 1, 3, 5, 15, 17, 25, 51, 75, 85, 255, 425, 1275
  • Factor pairs: 1275 = 1 × 1275, 3 × 425, 5 × 255, 15 × 85, 17 × 75, or 25 × 51,
  • Taking the factor pair with the largest square number factor, we get √1275 = (√25)(√51) = 5√51 ≈ 35.70714

1275 is the hypotenuse of SEVEN Pythagorean triples:
195-1260-1275 which is 15 times (13-84-85)
261-1248-1275 which is 3 times (87-416-425)
357-1224-1275 which is (7-24-25) times 51
540-1155-1275 which is 15 times (36-77-85)
600-1125-1275 which is (8-15-17) times 75
765-1020-1275 which is (3-4-5) times 255
891-912-1275 which is 3 times (297-304-425)

1274 Imagining With Sain Smart Jr.’s Tetris Puzzle

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My granddaughter recently received a Sain Smart Jr. Tetris Puzzle for her birthday. I got to watch as she and her younger sister had lots of fun creating pictures using the colorful pieces. Here are some of their creations:

 

As they played they experimented and learned what they liked and what they didn’t like. In the process, they learned some mathematics and may not have even realized it.

Yeah, they could also make all the Tetris puzzle pieces fit in the frame. It seems to help if you save some of the five wood grained pieces for last.

From what I could tell, this was a very fun and educational toy.

Now here I’ll share some information about the number 1274:

  • 1274 is a composite number.
  • Prime factorization: 1274 = 2 × 7 × 7 × 13, which can be written 1274 = 2 × 7² × 13
  • 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 × 3 × 2 = 12. Therefore 1274 has exactly 12 factors.
  • Factors of 1274: 1, 2, 7, 13, 14, 26, 49, 91, 98, 182, 637, 1274
  • Factor pairs: 1274 = 1 × 1274, 2 × 637, 7 × 182, 13 × 98, 14 × 91, or 26 × 49,
  • Taking the factor pair with the largest square number factor, we get √1274 = (√49)(√26) = 7√26 ≈ 35.69314

35² + 7² = 1274

1274 is the hypotenuse of a Pythagorean triple:
490-1176-1274 which is (5-12-13) times 98.
It is also 2(35)( 7), 35² – 7², 35² + 7²

1266 is a Centered Pentagonal Number

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1266 is the 23rd centered pentagonal number because 5(22)(23)/2 + 1 = 1266. The graphic below shows 1266 tiny dots arranged into a pentagonal shape and that 1266 is one more than five times the 22nd triangular number.

Here are some more facts about the number 1266:

  • 1266 is a composite number.
  • Prime factorization: 1266 = 2 × 3 × 211
  • 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 × 2 × 2 = 8. Therefore 1266 has exactly 8 factors.
  • Factors of 1266: 1, 2, 3, 6, 211, 422, 633, 1266
  • Factor pairs: 1266 = 1 × 1266, 2 × 633, 3 × 422, or 6 × 211
  • 1266 has no square factors that allow its square root to be simplified. √1266 ≈ 35.58089

1266 is also the sum of ten consecutive prime numbers:
103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 = 1266

 

1261 Can You Make a Star out of a Hexagon?

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Can you make a star out of a hexagon? If you have 1261 tiny squares arranged as a centered hexagon, you can rearrange those 1261 tiny squares into a six-pointed star as illustrated below!

37 was the last centered hexagonal number that was also a star number.

Here are some more facts about the number 1261:

  • 1261 is a composite number.
  • Prime factorization: 1261 = 13 × 97
  • 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 1261 has exactly 4 factors.
  • Factors of 1261: 1, 13, 97, 1261
  • Factor pairs: 1261 = 1 × 1261 or 13 × 97
  • 1261 has no square factors that allow its square root to be simplified. √1261 ≈ 35.51056

1261 is the sum of two squares two different ways:
30² + 19² = 1261
35² + 6² = 1261

1261 is the hypotenuse of FOUR Pythagorean triples:
420-1189-1261 calculated from 2(35)(6), 35² – 6², 35² + 6²
485-1164-1261 which is (5-12-13) times 97
539-1140-1261 calculated from 30² – 19², 2(30)(19), 30² + 19²
845-936-1261 which is 13 times (65-72-97)

Haunted Forest with 1260 Factor Trees

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1260 is the smallest number with 36 factors. That’s a new record. (32 was the old record and was held by both 840 and 1080.)

Often when a number has a lot of factors, we will visit a forest of its factor trees. 1260 certainly deserves such a forest. Since it is just before Halloween, It happens to be a haunted forest. Do you dare to go into such a forest? These three trees are scary enough for me! However, there are MANY more factor trees in that haunted forest! Perhaps if you are brave, you can find some of those factor trees in the haunted forest yourself.

Here’s more about the number 1260:

  • 1260 is a composite number.
  • Prime factorization: 1260 = 2 × 2 × 3 × 3 × 5 × 7, which can be written 1260 = 2² × 3² × 5 × 7.
  • The exponents in the prime factorization are 2, 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) (1 + 1) = 3 × 3 × 2 × 2  = 36. Therefore 1260 has exactly 36 factors.
  • Factors of 1260: 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 28, 30, 35, 36, 42, 45, 60, 63, 70, 84, 90, 105, 126, 140, 180, 210, 252, 315, 420, 630, 1260
  • Factor pairs: 1260 = 1 × 1260, 2 × 630, 3 × 420, 4 × 315, 5 × 252, 6 × 210, 7 × 180, 9 × 140, 10 × 126, 12 × 105, 14 × 90, 15 × 84, 18 × 70, 20 × 63, 21 × 60, 28 × 45, 30 × 42 or 35 × 36
  • Taking the factor pair with the largest square number factor, we get √1260 = (√36)(√35) = 6√35 ≈ 35.49648

21 × 60 = 1260 The same digits are used on both sides of that equation and that makes 1260 the 19th Friedman number.

1260 is also the sum of the interior angles of a nine-sided polygon. Convex or concave, that is the sum. The concave nonagon below is a good illustration of that fact:

1260 is also the hypotenuse of a Pythagorean triple:
756-1008-1260 which is (3-4-5) times 252

1252 What Do You Notice?

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Remembering that 1250 = 25² + 25² from just a couple of days ago, I was struck when I noticed that 1252 = 24² + 26². I wondered if it was part of a pattern, so I made this chart. What do you think?

When I thought about it more, I realized that perhaps it isn’t so remarkable. After all,
(n – 1)² + (n + 1)² = (n² – 2n +1) + (n² + 2n +1) = n² + n² – 2n + 2n + 1 + 1  = 2n² + 2
Nevertheless, I still like the pattern.

Here are some more facts about the number 1252:

  • 1252 is a composite number.
  • Prime factorization: 1252 = 2 × 2 × 313, which can be written 1252 = 2² × 313
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 1252 has exactly 6 factors.
  • Factors of 1252: 1, 2, 4, 313, 626, 1252
  • Factor pairs: 1252 = 1 × 1252, 2 × 626, or 4 × 313
  • Taking the factor pair with the largest square number factor, we get √1252 = (√4)(√313) = 2√313 ≈ 35.38361

Because 1252 = 26² + 24², it is the hypotenuse of a Pythagorean triple:
100-1248-1252 calculated from 26² – 24², 2(26)(24), 26² + 24²

100-1248-1252 is also 4 times (25-312-313)

 

1247 Is a Pentagonal Number

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Two factors of 1247 make it the 29th pentagonal number. Here’s why:

29(3·29-1)/2 = 29(86)/2 = 29(43) = 1247

Here is an illustration of this pentagonal number featuring a different, but equivalent, formula.  Seeing the pentagonal numbers less than 1247 in the illustration won’t be difficult either.

Here are some more facts about the number 1247:

  • 1247 is a composite number.
  • Prime factorization: 1247 = 29 × 43
  • 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 1247 has exactly 4 factors.
  • Factors of 1247: 1, 29, 43, 1247
  • Factor pairs: 1247 = 1 × 1247 or 29 × 43
  • 1247 has no square factors that allow its square root to be simplified. √1247 ≈ 35.31289

1247 is the sum of consecutive prime numbers two different ways:
It is the sum of the twenty-three prime numbers from 11 to 103.
It is also the sum of seven consecutive primes:
163 + 167 + 173 + 179 + 181 + 191 + 193 = 1247

1247 is the hypotenuse of a Pythagorean triple:
860-903-1247 which is (20-21-29) times 43

 

1242 is a Decagonal Number

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If you had 1242 tiny little squares you could arrange them into a decagon, just as I did for the graphic below.

18 is a factor of 1242. Since 18(4·18-3) = 18(69) = 1242, it is the 18th decagonal number.

Here are a few more facts about the number 1242:

  • 1242 is a composite number.
  • Prime factorization: 1242 = 2 × 3 × 3 × 3 × 23, which can be written 1242 = 2 × 3³ × 23
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 × 2 × 2 = 16. Therefore 1242 has exactly 16 factors.
  • Factors of 1242: 1, 2, 3, 6, 9, 18, 23, 27, 46, 54, 69, 138, 207, 414, 621, 1242
  • Factor pairs: 1242 = 1 × 1242, 2 × 621, 3 × 414, 6 × 207, 9 × 138, 18 × 69, 23 × 54, or 27 × 46
  • Taking the factor pair with the largest square number factor, we get √1242 = (√9)(√138) = 3√138 ≈ 35.24202

1242 is the sum of consecutive prime numbers two different ways:
It is the sum of the eighteen prime numbers from 31 to 107, and
it is also the sum of the sixteen prime numbers from 43 to 109.