## Sunday, May 21, 2017

### The race of Cloud Computing

I wrote earlier about a news digest app returning supplemental information which is not directly related to the article. Today, I read on the app that the horse "Cloud Computing" has won the Preakness race. And sure enough, there is a box underneath the article with a definition of cloud computing from Wikipedia. What is ironic is that this news summary is probably enabled by leveraging cloud computing. "Cloud Computing" won over "Classic Empire" by coming in from behind and won the race by a head. Perhaps this is an indication how cloud computing will overtake the classical empire of on premise computing😊.

## Thursday, May 11, 2017

### Find interesting properties of your favorite number

I created a simple web application where you can enter a positive integer < 10

To start, please visit: http://postvakje.pythonanywhere.com/

^{12 }, wait a few moments and it will return a list of interesting properties about this number. Please give it a try and let me know what you think!To start, please visit: http://postvakje.pythonanywhere.com/

## Sunday, May 7, 2017

## Tuesday, May 2, 2017

### 12345

The number 12345 is interesting as it is a sphenic number (product of 3 distinct primes) and its digit reversal 54321 is also a sphenic number.

## Tuesday, April 18, 2017

### Palindromic primes which are sums of 3 consecutive primes

OEIS sequence A113846 lists palindromic primes $q$ which are the sums of 3 consecutive primes $p_1$, $p_2$, $p_3$ such that $p_2$ is also a palindromic prime.

The first digit of $p_2$ cannot be 2 since $p_2$ is odd. Next we show that the first digit of $p_2$ is either 1 or 3.

Let $k$ be an even integer such that $p_2$ has $k+1$ digits. Suppose the first digit of $p_2$ is 4 or larger, i.e. $4\cdot 10^k \leq p_2 < 10^{k+1}$. Then by Bertrand's postulate (or Bertrand-Chebyshev theorem), $p_1\geq \frac{p_2}{2}$ and $p_3 \leq 2p_2$. This implies that $10^{k+1} > p_1 \geq 2\cdot 10^k$, $4\cdot 10^k < p3 \leq 2\cdot 10^{k+1}$ and thus $10^{k+1} < q < 4\cdot 10^{k+1}$, i.e. $q$ has an even number of digits, a contradiction.

Next we show that the first digit of $q$ is 3 or 9. To do this, we need a stronger result than Bertrand's postulate for the prime gap. Since $p_2 > 647$, Rohrback and Weis's 1964 result [1] shows that $p_1 \geq \frac{12}{13}p_2$ and $p_3 \leq \frac{14}{13}p_2$.

This shows that $2\frac{12}{13} p_2 \leq q \leq 3\frac{1}{13} p_2$. Since $q$ is prime, it cannot start with the digit 5. If $p_2$ start with the digit 1, then $2\cdot 10^k \leq q \leq 6 \frac{2}{13} 10^k$ and thus $q$ must start with the digit 3.

If $p_2$ start with the digit 3, then $8\frac{10}{13}10^k \leq q \leq 12\frac{4}{13}10^k$. Since $q < 10^{k+1}$, this means that $q$ must start with the digit 9.

Clearly $p_3$ must have the same number of digits as $p_2$ and as $q$.

Finally, we show that $p_3$ has the same starting digit as $p_2$. Suppose $p_2$ starts with the digit 1. If $p_3$ has a different starting digit, this must mean that $p_3 \geq 2\cdot 10^k$, i.e.

$p_2 \geq 1\frac{11}{13}10^k$, $p_1\geq 1\frac{119}{169} 10^k$ and $q \geq 4\cdot 10^k$, a contradiction.

Suppose $p_2$ start with the digit 3. If $p_3$ has a different starting digit, this must mean that $p_3 \geq 4\cdot 10^k$, i.e.

$p_2 \geq 3\frac{9}{13} 10^k$, $p_1 \geq 3\frac{69}{169}10^k$ and $q\geq 10^{k+1}$, a contradiction. QED

This argument also shows that to search for terms, for each $k$, one only needs to consider for $p_2$ the palindromic primes in the ranges $(10^k, 1\frac{7}{19}10^k)$ and $(3\cdot 10^k, 3 \frac{8}{19}\cdot 10^k)$ which amount to approximately $0.79\cdot 10^{\frac{k}{2}}$ palindromes to consider rather than a full search of $10\cdot 10^{\frac{k}{2}}$ palindromes of length $k+1$.

1. H. Rohrback and J. Weis, "Zum finiten Fall des Bertrandschen Postulats", J. Reine Angew. Math., 214/215 (1964), 432–440.

**Theorem**: All terms of A113846 have an odd number of digits and the first digit (and last digit) is either 3 or 9. In addition, $p_2$ and $p_3$ have the same first digit and $p_2$ and $p_3$ have the same number of digits as $q$. The first digit of $p_2$ (and $p_3$) is equal to the first digit of $q$ divided by 3, i.e. if the first digit of $q$ is $3$, then the first digit of $p_2$ and $p_3$ is $1$.**Proof**: First note that all palindromes with an even number of digits is divisible by 11. Let $p_1$, $p_2$, $p_3$ be 3 consecutive primes such that $p_2$ is a palindrome and $q = p_1+p_2+p_3$ is a palindromic prime. It is clear that $q, p_2 > 11$ and have a odd number of digits since the first term of A113846 is 31513.The first digit of $p_2$ cannot be 2 since $p_2$ is odd. Next we show that the first digit of $p_2$ is either 1 or 3.

Let $k$ be an even integer such that $p_2$ has $k+1$ digits. Suppose the first digit of $p_2$ is 4 or larger, i.e. $4\cdot 10^k \leq p_2 < 10^{k+1}$. Then by Bertrand's postulate (or Bertrand-Chebyshev theorem), $p_1\geq \frac{p_2}{2}$ and $p_3 \leq 2p_2$. This implies that $10^{k+1} > p_1 \geq 2\cdot 10^k$, $4\cdot 10^k < p3 \leq 2\cdot 10^{k+1}$ and thus $10^{k+1} < q < 4\cdot 10^{k+1}$, i.e. $q$ has an even number of digits, a contradiction.

Next we show that the first digit of $q$ is 3 or 9. To do this, we need a stronger result than Bertrand's postulate for the prime gap. Since $p_2 > 647$, Rohrback and Weis's 1964 result [1] shows that $p_1 \geq \frac{12}{13}p_2$ and $p_3 \leq \frac{14}{13}p_2$.

This shows that $2\frac{12}{13} p_2 \leq q \leq 3\frac{1}{13} p_2$. Since $q$ is prime, it cannot start with the digit 5. If $p_2$ start with the digit 1, then $2\cdot 10^k \leq q \leq 6 \frac{2}{13} 10^k$ and thus $q$ must start with the digit 3.

If $p_2$ start with the digit 3, then $8\frac{10}{13}10^k \leq q \leq 12\frac{4}{13}10^k$. Since $q < 10^{k+1}$, this means that $q$ must start with the digit 9.

Clearly $p_3$ must have the same number of digits as $p_2$ and as $q$.

Finally, we show that $p_3$ has the same starting digit as $p_2$. Suppose $p_2$ starts with the digit 1. If $p_3$ has a different starting digit, this must mean that $p_3 \geq 2\cdot 10^k$, i.e.

$p_2 \geq 1\frac{11}{13}10^k$, $p_1\geq 1\frac{119}{169} 10^k$ and $q \geq 4\cdot 10^k$, a contradiction.

Suppose $p_2$ start with the digit 3. If $p_3$ has a different starting digit, this must mean that $p_3 \geq 4\cdot 10^k$, i.e.

$p_2 \geq 3\frac{9}{13} 10^k$, $p_1 \geq 3\frac{69}{169}10^k$ and $q\geq 10^{k+1}$, a contradiction. QED

This argument also shows that to search for terms, for each $k$, one only needs to consider for $p_2$ the palindromic primes in the ranges $(10^k, 1\frac{7}{19}10^k)$ and $(3\cdot 10^k, 3 \frac{8}{19}\cdot 10^k)$ which amount to approximately $0.79\cdot 10^{\frac{k}{2}}$ palindromes to consider rather than a full search of $10\cdot 10^{\frac{k}{2}}$ palindromes of length $k+1$.

**Conjecture**: For all terms in A113846, the corresponding primes $p_1$ has the same number of digits and the same first digit as $p_2$ and $p_3$.**References:**1. H. Rohrback and J. Weis, "Zum finiten Fall des Bertrandschen Postulats", J. Reine Angew. Math., 214/215 (1964), 432–440.

## Thursday, March 30, 2017

### Spring is here

It is amazing how a couple of days of sunshine and warmer weather erased most of the evidence that we had a major winter storm 2 weeks ago.

## Wednesday, March 15, 2017

## Tuesday, March 14, 2017

### Pi Day 2017

Happy Pi day (and a snowy one in the Northeastern US)! One of the early childhood mathematical memories I had was learning that π can be approximated by 22/7 and accurate up to 2 decimal places, a fact that was known since antiquity. In particular, Archimedes gave the first proof that 22/7 is strictly larger than π. The fifth century Chinese mathematician and astronomer Zu Chongzhi gave the approximation 355/113 that is accurate up to 6 decimal places. Today, sophisticated algorithms using Ramanujan's formula and its variants can compute trillions of digits of π.

## Thursday, March 9, 2017

### Binomial coefficients

When studying binomial coefficients and how they form Pascal's Triangle, we learn that they are the coefficients of $(x+y)^n$, i.e.

$$(x+y)^n = \sum_{k=0}^n \left(\begin{array}{c}n\\k\end{array}\right)x^ky^{n-k}$$

Replacing $y$ with $-y$ we get:

$$(x-y)^n = \sum_{k=0}^n -1^k\left(\begin{array}{c}n\\k\end{array}\right)x^{n-k}y^{k}$$

For example, the coefficients of $(x+y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$ are 1,4,6,4,1 which form the 5th row of Pascal's triangle.

What happens if the coefficients of the polynomial are repeated binomial coefficients such as 1,1,4,4,6,6,4,4,1,1? What is $$\sum_{k=0}^{2n+1} \left(\begin{array}{c}n\\\lfloor k/2\rfloor\end{array}\right)x^ky^{n-k}?$$

It is easy to show that this is equal to $y(x^2+y^2)^n + x(x^2+y^2)^n = (x+y)(x^2+y^2)^n$.

What about repeating coefficients with sign changes such as 1,1,-4,-4,6,6,-4,-4,1,1?

The same argument shows that

$$\sum_{k=0}^{2n+1} -1^{\lfloor k/2\rfloor}\left(\begin{array}{c}n\\\lfloor k/2\rfloor\end{array}\right)x^{n-k}y^{k}$$

is equal to $y(x^2-y^2)^n + x(x^2-y^2)^n = (x+y)(x^2-y^2)^n$ which can be further simplified as $(x+y)^{n+1}(x-y)^n$.

If we now change the sign of every other coefficient as well, i.e. 1,-1,-4,4,6,-6,-4,4,1, -1,

we get

$$\sum_{k=0}^{2n+1} -1^{\lfloor k/2\rfloor+n-k}\left(\begin{array}{c}n\\\lfloor k/2\rfloor\end{array}\right)x^{n-k}y^{k}$$

This can be shown to be equal to $x(x^2-y^2)^n -y(x^2-y^2)^n = (x-y)(x^2-y^2)^n$ which is equal to $(x-y)^{n+1}(x+y)^n$.

We can repeat the coefficients more than 2 times as well and show that

$$\sum_{k=0}^{m(n+1)-1} \left(\begin{array}{c}n\\\lfloor k/m\rfloor\end{array}\right)x^ky^{n-k}$$

is equal to $\sum_{t=0}^{t=m-1}x^ty^{m-1-t}(x^m+y^m)^n$.

## Friday, January 20, 2017

### Interesting number: 1440000

The number 1440000 satisfies the following equation:

$$1440000^3 - 1 = 1439940^3 + 71999^3 = 1440060^3 - 72001^3$$

By Fermat's Last Theorem (for which the case $n=3$ was proven by Euler), $1440000^3$ cannot be the sum or difference of 2 positive integer cubes. The above equation shows that $1440000^3-1$ is the sum and the difference of 2 positive integer cubes.

$$1440000^3 - 1 = 1439940^3 + 71999^3 = 1440060^3 - 72001^3$$

By Fermat's Last Theorem (for which the case $n=3$ was proven by Euler), $1440000^3$ cannot be the sum or difference of 2 positive integer cubes. The above equation shows that $1440000^3-1$ is the sum and the difference of 2 positive integer cubes.

Subscribe to:
Posts (Atom)