Carmichael number

1729 Not a Dull Number

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Today’s Puzzle:

A little more than a hundred years ago near Cambridge University G. H. Hardy took a taxi to visit his young friend and fellow mathematician, Srinivasa Ramanujan, in the hospital. Hardy couldn’t think of anything interesting about his taxi number, 1729, and remarked to Ramanujan that it appeared to be a rather dull number. But even the reason for his hospitalization could not prevent Ramanujan’s genius from shining through. He immediately recognized 1729’s unique and very interesting attribute: it is the SMALLEST number that can be written as the sum of two cubes in two different ways! Indeed,
12³ + 1³ = 1729, and
10³ + 9³ = 1729.

Today’s puzzle looks a little bit like a modern-day American taxi cab with the clues 17 and 29 at the top of the cab. The table below the puzzle contains all the Pythagorean triples with hypotenuses less than 100 sorted by legs and by hypotenuses. Use the table and logic to write the missing sides of the triangles in the puzzle. The right angle on each triangle is the only one that is marked. Obviously, none of the triangles are drawn to scale. No Pythagorean triple will appear more than once in the puzzle.

Sorted TriplesHere’s the same puzzle without all the added color:

Print the puzzles or type the solutions in this excel file: 10 Factors 1721-1729.

What taxi cab might Hardy have tried to catch next? He might have had to wait a long time for it, 4104.
16³ + 2³ = 4104, and
15³ + 9³ = 4104.

Factors of 1729:

  • 1729 is a composite number.
  • Prime factorization: 1729 = 7 × 13 × 19.
  • 1729 has no exponents greater than 1 in its prime factorization, so √1729 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 1729 has exactly 8 factors.
  • The factors of 1729 are outlined with their factor pair partners in the graphic below.

More About the Number 1729:

Did you notice these cool-looking factor pairs?
19 · 91 = 1729.
13 · 133 = 1729.

1729 is the hypotenuse of a Pythagorean triple:
665-1596-1729 which is 133 times (5-12-13).

1729 is the difference of two squares in four different ways:
865² – 864² = 1729,
127² – 120² = 1729,
73² – 60² = 1729, and
55² – 36² = 1729.

1729 is also a pseudoprime number! For example, even though it is a composite number, it passes the quick prime number test I first wrote about in 341 is the smallest composite number that gives a false positive for this Quick Prime Number Test:

It is known as a Carmichael number because back in 1909, Robert D. Carmichael, an American mathematician, showed that it passed even more prime number tests than this one.

Mathemagical Properties of 1105

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1105 is the magic sum of a 13 × 13 magic square. Why?
Because 13×13 = 169 and 169×170÷2÷13 = 13×85 = 1105.

If you follow the location of the numbers 1, 2, 3, 4, all the way to 169 in the magic square, you will see the pattern that I used to make that magic square. If you click on 10-factors-1102-1110  and go to the magic squares tab, you can use the same pattern or try another to create an 11 × 11, 13 × 13, or 15 × 15 magic square. The sums on the rows, columns, and diagonals will automatically populate as you write in the numbers so you can verify that you have indeed created a magic square.

1105 tiny squares can be made into a decagon so we say it is a decagonal number:

Those 1105  tiny squares can also be arranged into a centered square:

Why is 1105 the 24th Centered Square Number? Because it is the sum of consecutive square numbers:
24² + 23² = 1105

But that’s not all! 1105 is the smallest number that is the sum of two squares FOUR different ways:

24² + 23² = 1105
31² + 12² = 1105
32² + 9² = 1105
33² + 4² = 1105

1105 is also the smallest number that is the hypotenuse of THIRTEEN different Pythagorean triples. Yes, THIRTEEN! (Seven was the most any previous number has had.) It is also the smallest number to have FOUR of its Pythagorean triplets be primitives (Those four are in blue type.):

47-1104-1105 calculated from 24² – 23², 2(24)(23), 24² + 23²
105-1100-1105 which is 5 times (21-220-221)
169-1092-1105 which is 13 times (13-84-85)
264-1073-1105 calculated from 2(33)(4), 33² – 4², 33² + 4²
272-1071-1105 which is 17 times (16-63-65)
425-1020-1105 which is (5-12-13) times 85
468-1001-1105 which is 13 times (36-77-85)
520-975-1105 which is (8-15-17) times 65
561-952-1105 which is 17 times (33-56-85)
576-943-1105 calculated from 2(32)(9), 32² – 9², 32² + 9²
663-884-1105 which is (3-4-5) times 221
700-855-1105 which is 5 times (140-171-221)
744-817-1105 calculated from 2(31)(12), 31² – 12², 31² + 12²

Why is it the hypotenuse more often than any previous number? Because of its factors! 1105 = 5 × 13 × 17, so it is the smallest number that is the product of THREE different Pythagorean hypotenuses.

It gets 1 triple for each of its three individual factors: 5, 13, 17, 2 triples for each of the three ways the factors can pair up with each other: 65, 85, 221, and four primitive triples for the one way they can all three be together: 1105. Thus it gets 2º×3 + 2¹×3 + 2²×1 = 3 + 6 + 4 = 13 triples.

Speaking of factors, let’s take a look at 1105’s factoring information:

  • 1105 is a composite number.
  • Prime factorization: 1105 = 5 × 13 × 17
  • 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 1105 has exactly 8 factors.
  • Factors of 1105: 1, 5, 13, 17, 65, 85, 221, 1105
  • Factor pairs: 1105 = 1 × 1105, 5 × 221, 13 × 85, or 17 × 65
  • 1105 has no square factors that allow its square root to be simplified. √1105 ≈ 33.24154

1105 is also a palindrome in four different bases, and I also like the way it looks in base 8:
It’s 10001010001 in BASE 2 because 2¹º + 2⁶ + 2⁴ + 2º = 1105,
101101 in BASE 4 because 4⁵ + 4³ + 4² + 4º = 1105,
2121 in BASE 8 because 2(8³) + 1(8²) + 2(8) + 1(1) = 1105,
313 in BASE 19 because 3(19²) + 1(19) + 3(1) = 1105
1M1 in BASE 24 (M is 22 base 10) because 24² + 22(24) + 1 = 1105

Last, but certainly not least, you wouldn’t think 1105 is a prime number, but it is a pseudoprime: the second smallest Carmichael number. Only Carmichael number 561 is smaller than it is.

A Carmichael number is a composite number that behaves like a prime number by giving a false positive to all of certain quick prime number tests:
1105 passes the test p¹¹⁰⁵ Mod 1105 = p for all prime numbers p < 1105. Here is an image of my computer calculator showing 1105 passing the first five tests! Only a prime number should pass all these tests.

1105 is indeed a number with amazing mathemagical properties!

561 gives a false positive to these 102 prime number tests

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We can use some easy divisibility tests to find two of the factors of 561.

  • 5 + 6 + 1 = 12, a multiple of 3 so 561 is divisible by 3.
  • 5 – 6 + 1 = 0, which is divisible by 11 so 561 can be evenly divided by 11.

But if we only do those divisibility tests, we will miss something very significant about the number 561:

If you divide 2^561 by 561, the remainder will be 2. When we are more interested in the remainder than the quotient, we can simply type “2, x^y, 561, Mod, 561, =” into the computer’s scientific calculator:

561 Mod Calculator

This is only a picture of a calculator.

2^561 (mod 561) = 2 means that 561 is VERY LIKELY a prime number, but this is one time when VERY LIKELY does not mean ACTUALLY!

561 has something in common with the number 341. Yes, both of them pass this quick prime number test, and both of them are composite numbers divisible by 11. Both numbers are called pseudo-prime numbers. (341 and 561 are the two smallest composite numbers to give a false positive to this particular test.)

561 is even more remarkable than 341:

  • 2^561 (mod 561) = 2, and 2^341 (mod 341) = 2 (Both numbers pass.)
  • 3^561 (mod 561) = 3, while 3^341 (mod 341) = 168 (561 passes; 341 fails.)
  • 5^561 (mod 561) = 5
  • 7^561 (mod 561) = 7
  • 11^561 (mod 561) = 11
  • 13^561 (mod 561) = 13
  • 17^561 (mod 561) = 17
  • etc.

There are 102 prime numbers less than 561, and p^561 (mod 561) = p for every single one of them! 561 acts like a prime number in those 102 ways.

In 1910 R. D. Carmichael discovered that 561 is the first COMPOSITE number that passes ALL those modular (remainder) prime number tests, so 561 is the first Carmichael number. Yes, there will be more – in fact, infinitely more.

R. D. Carmichael actually found that 561 passes ALL 559 prime number tests using each whole number between 1 and 561, for example 33^561 (mod 561) = 33. All prime numbers can make a similar claim, but 561 is the smallest composite number with that property.

(Note: I did not use the standard mathematical notation for this property, but what I used is equivalent to it and doesn’t require parenthesis when typing it into the computer’s scientific calculator. Also I think “=” is less intimidating looking for some of my readers than “≡”.)

There are other reasons why the number 561 is an interesting number:

Because 33 x 34/2 = 561, we know that 561 is the 33rd triangular number and is equal to 1 + 2 + 3 + . . . + 31 + 32 + 33, the sum of the first 33 whole numbers.

Because 17 x (2 x 17 – 1) = 561, we know that 561 is the 17th hexagonal number. (All hexagonal numbers are also triangular numbers.)

561 is also the hypotenuse of the Pythagorean triple 264-495-561. What is the greatest common factor of those three numbers? Hint: it is one of the factors of 561 listed below:

——————————————————————–

  • 561 is a composite number.
  • Prime factorization: 561 = 3 x 11 x 17
  • 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 561 has exactly 8 factors.
  • Factors of 561: 1, 3, 11, 17, 33, 51, 187, 561
  • Factor pairs: 561 = 1 x 561, 3 x 187, 11 x 51, or 17 x 33
  • 561 has no square factors that allow its square root to be simplified. √561 ≈ 23.6854