In my previous post I suggested that there are products of primes p and q that are easier (relatively) to factor because there is an integer k such that k = (p*q - 1)/2 and k^k = (phi(p*q))/2 and I called k a selfie power.
Before I talk about such products, I'll talk about products of primes, which have no selfie power and no semi selfie powers* because proper categorization is important.
Theorem: Given a product of primes p and q, if p or q divides phi(p*q) then p*q has no selfie power (ie an integer k such that k = (p*q - 1)/2 and k^k = (phi(p*q))/2 )
Proof: If p or q divides phi(p*q) then GCD(phi(p*q), k) = 1 since by definition of a selfie power k = (p*q - 1)/2 but also by definition of a selfie power k^k = (phi(p*q))/2 so 2*(k^k) = phi(p*q), which is a contradiction.
Example: Let p = 5 and q = 11, then p*q = 55 and phi(p*q) = (p-1)*(q-1) = 4*10 = 40 and clearly p divides phi(p*q). Furthermore, (p*q-1)/2 = 27 and 27^27 (mod 55) = 42 so 27 is not a selfie square as expected.
*to be defined in future blog posts
The Bad Byte
4.9.17
30.8.17
Selfie Powers
According to some people in Israel, the number 36 is special. According to these people, at any point in time there are 36 people who hold the world together and if one of them dies, then the world will fall apart. I don’t share such esoteric views but I do find the number 35, which is one less than 36, to be quite extraordinary from a mathematical point of view.
The exponentiation table of Z sub 35 |
It represents a special class of products of two distinct primes with peculiar properties. Such products of primes are really easy to factor.
Before I talk more about other products of primes like this, first I have to introduce a new definition, which is not new to me but it is new to this blog.
Definition: Selfie powers are integers k mod n such that 2*k = n-1 and k^k = phi(n)/2 where phi(n) is Euler's Totient Function
Example:
Given n = 35 then (n-1)/2 = 17 and 17^17 = 12, which is phi(35)/2
25.7.17
A Few More Conjectures
The subset of safe primes, which I lovingly call "steady" primes are not so steady after all if my latest
conjectures turn out to be true.
As I wrote in a previous post, the set of safe primes is composed of two subsets: the set of steady primes and the set of tough primes.
Conjecture 1: Given p a steady prime and a < p such that it is not a power of 2 mod p then
Conjecture 2: Given p a steady prime then (p-1)/2 is not a power of 2 mod p and
conjectures turn out to be true.
As I wrote in a previous post, the set of safe primes is composed of two subsets: the set of steady primes and the set of tough primes.
The first few safe primes where primes in olive green are steady and the rest are tough |
Conjecture 1: Given p a steady prime and a < p such that it is not a power of 2 mod p then
a^((p-1)/2)= p-1 else if a is a power of 2 mod p then a^((p-1)/2)= 1
Conjecture 2: Given p a steady prime then (p-1)/2 is not a power of 2 mod p and
((p-1)/2)^((p-1)/2) = p-1
Labels:
Mathematics
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safe primes
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steady primes
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tough primes
13.6.17
2.6.17
The Graphing Calculator
When I first moved to Canada I was simultaneously shocked and pleasantly relieved to find out that calculators are mandatory in high schools.
26.4.17
Subsets of Safe Primes
The set of safe primes is composed of 2 mutually disjoint sets: the set of tough primes and the set of steady primes.
The main difference between these two sets lies in the structure of the exponentiation tables of their elements. For a tough prime p, I conjecture that:
Conjecture: If p is a prime of the form 8n+7, then the order of even powers of 2 mod p is p-1 and the order of odd powers of 2 mod p is (p-1)/2
In contrast, for a steady prime q I conjecture that:
Conjecture: If q is of the form 8n-1 such that 4n-1 and 8n-1 are also primes, then the order of all powers of 2 mod q is equal to (p-1)/2
Example: Let p = 11 and let q = 7. Below are the corresponding exponentiation tables of Z_{11} and Z_{7}.
The order of 2 mod 11 = 10, which is equal to the order of 8 mod 11, but the order of 4 mod 11 is half of that. Whereas in Z_{7} the order of all powers of 2 is equal to 3.
Steady primes are the only primes where the order of all powers of 2 mod p is the same. For all other primes the order of different powers of 2 is different.
On a slightly different note, as bad as safe primes may sound for cryptography, they are still not as bad as strong primes.
a list of the first few safe primes with tough primes in light blue background; the rest are steady primes |
Conjecture: If p is a prime of the form 8n+7, then the order of even powers of 2 mod p is p-1 and the order of odd powers of 2 mod p is (p-1)/2
In contrast, for a steady prime q I conjecture that:
Conjecture: If q is of the form 8n-1 such that 4n-1 and 8n-1 are also primes, then the order of all powers of 2 mod q is equal to (p-1)/2
Example: Let p = 11 and let q = 7. Below are the corresponding exponentiation tables of Z_{11} and Z_{7}.
The order of 2 mod 11 = 10, which is equal to the order of 8 mod 11, but the order of 4 mod 11 is half of that. Whereas in Z_{7} the order of all powers of 2 is equal to 3.
Steady primes are the only primes where the order of all powers of 2 mod p is the same. For all other primes the order of different powers of 2 is different.
On a slightly different note, as bad as safe primes may sound for cryptography, they are still not as bad as strong primes.
Labels:
Mathematics
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Order of An Element
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RSA Problem
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safe primes
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steady primes
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tough primes
20.4.17
Why Steady Primes?
In my previous entry I discussed a subset of safe primes with an interesting property.
It appears that when a prime p is of the form 8k - 1 where 4k - 1 and 8k - 1 are both primes, then the order of each power of 2 mod p is phi(p)/2 where phi() is the Euler's Totient Function.
This is only one part of my conjecture. Here's another part of it:
Conjecture: If p is a prime of the form 8k - 1 where 4k - 1 and 8k - 1 are both primes and n is an integer smaller than q such that it is not a power of 2 mod p, then the order of n mod p is phi(p)= p-1
Example: Below is an image of the exponentiation tables of Z_{23} and Z_{7}.
In my previous entry I called such primes "steady primes". The reason why I chose the name "steady" is because of the "steady", uniform structure of the exponentiation table of Z_{p} when p is a steady prime.
All elements of Z_{p} when p is a steady prime have order that is equal to or greater than phi(p)/2. This is also true for the rest of the safe primes, which are the tough primes, but the structure of the exponentiation table of Z_{p} when p is a tough prime is different. With tough primes every odd power of 2 mod p has order p-1 and every even power of 2 mod p has order phi(p)/2.
Conjecture: If p is a steady prime, then phi(p)/2 is also a prime number.
Example: The first few steady primes are: 7,23,47,167,263,359,383,479,503
phi(p)/2 when p is the first few steady primes is equal to: 3,11, 23, 83, 131, 179, 191, 239, 261
Unfortunately, the curious properties of steady primes are not applicable to all products of steady primes. There are some products of steady primes pq with such uniform structure of the exponentiation table of Z_{pq} but they are kind of rare.
Below is a calculator that can be used to generate powers of powers of integers modulo other integers, the algorithm for which I described here.
It appears that when a prime p is of the form 8k - 1 where 4k - 1 and 8k - 1 are both primes, then the order of each power of 2 mod p is phi(p)/2 where phi() is the Euler's Totient Function.
This is only one part of my conjecture. Here's another part of it:
Conjecture: If p is a prime of the form 8k - 1 where 4k - 1 and 8k - 1 are both primes and n is an integer smaller than q such that it is not a power of 2 mod p, then the order of n mod p is phi(p)= p-1
Example: Below is an image of the exponentiation tables of Z_{23} and Z_{7}.
In my previous entry I called such primes "steady primes". The reason why I chose the name "steady" is because of the "steady", uniform structure of the exponentiation table of Z_{p} when p is a steady prime.
All elements of Z_{p} when p is a steady prime have order that is equal to or greater than phi(p)/2. This is also true for the rest of the safe primes, which are the tough primes, but the structure of the exponentiation table of Z_{p} when p is a tough prime is different. With tough primes every odd power of 2 mod p has order p-1 and every even power of 2 mod p has order phi(p)/2.
Conjecture: If p is a steady prime, then phi(p)/2 is also a prime number.
Example: The first few steady primes are: 7,23,47,167,263,359,383,479,503
phi(p)/2 when p is the first few steady primes is equal to: 3,11, 23, 83, 131, 179, 191, 239, 261
Unfortunately, the curious properties of steady primes are not applicable to all products of steady primes. There are some products of steady primes pq with such uniform structure of the exponentiation table of Z_{pq} but they are kind of rare.
Below is a calculator that can be used to generate powers of powers of integers modulo other integers, the algorithm for which I described here.
Labels:
Euler's Phi Function
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Euler's Totient Function
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Mathematics
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steady primes
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tough primes
Steady Primes
Another interesting subset of the set of safe primes is the set of steady primes as defined below:
Definition: A steady prime is a safe prime p such that all powers of 2 mod p have order phi(p)/2 where phi() is Euler's Totient Function.
Example: Let p = 23.
Using a calculator, it is easy to see that all unique powers of 2 mod 23 are 2,4,8,16,9,18,13,3,6,12,1
Using another calculator, it is also easy to verify that the order of 2 mod 23, or the smallest integer k for which 2^k = 1 mod 23 is 11, the order of 4 mod 23 is also 11, and so is the order of 8 mod 23, and this is the case for every other power of 2 mod 23 in the list above.
The first steady primes are 7,23,47,167,263,359,383,479,503,...
Conjecture: Steady primes are primes of the form 8p-1 such that 4p-1 and 8p-1 are also primes.
Claim: Given two steady primes p and q, the order of every power of 2 mod p*q is phi(p*q)/4
Claim: For all other safe primes q that are not steady primes, the order of at least one power of 2 mod q is at least 2 times less than the order of 2 mod q itself.
Below is a calculator for finding powers of any integer a mod n such that a < n.
Definition: A steady prime is a safe prime p such that all powers of 2 mod p have order phi(p)/2 where phi() is Euler's Totient Function.
Example: Let p = 23.
Using a calculator, it is easy to see that all unique powers of 2 mod 23 are 2,4,8,16,9,18,13,3,6,12,1
Using another calculator, it is also easy to verify that the order of 2 mod 23, or the smallest integer k for which 2^k = 1 mod 23 is 11, the order of 4 mod 23 is also 11, and so is the order of 8 mod 23, and this is the case for every other power of 2 mod 23 in the list above.
The first steady primes are 7,23,47,167,263,359,383,479,503,...
Conjecture: Steady primes are primes of the form 8p-1 such that 4p-1 and 8p-1 are also primes.
Claim: For all other safe primes q that are not steady primes, the order of at least one power of 2 mod q is at least 2 times less than the order of 2 mod q itself.
Below is a calculator for finding powers of any integer a mod n such that a < n.
Labels:
Euler's Phi Function
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Euler's Totient Function
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Mathematics
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safe primes
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steady primes
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tough primes
26.3.17
Other Collatz-like Algorithms
I wrote about the Collatz conjecture in relation to finite fields and I came across a few interesting sequences but I didn't discuss the possibility of other Collatz-like algorithms that always reach 1.
Once again, the Collatz problem is defined as follows:
Collatz Algorithm: Given any positive integer n, let a_{0} = n. For i > 0, let:
a_{i} = a_{i-1}/2 if a_{i-1} is even
a_{i} = 3*a_{i-1} + 1 if a_{i-1} is odd
Conjecture: Given any positive integer, the above recurrence relation returns 1.
This conjecture seems to hold although it hasn't been proven yet.
While looking at a different problem related to generating powers of 2 backwards, I stumbled upon a Collatz-like algorithm, which I generalize below for all positive integers.
Collatz-like Algorithm #1: Given any positive integer n, let a_{0} = (n+1)/2 if n is odd or a_{0} = n/2 if n is even. For i > 0, let:
a_{i} = a_{i-1}/2 if a_{i-1} is even
a_{i} = (a_{i-1} - 1)/2 + a_{0} if a_{i-1} is odd
Conjecture: Given any positive integer n, the above recurrence relation returns 1, and all integers it reaches before it reaches 1 are strictly less than n
In fact, I claim that:
Claim: If n is even then the Collatz-like Algorithm #1 generates all powers of 2 mod (n-1) in reverse order and if n is odd then it generates all powers of 2 mod n in reverse order.
Example: let n = 36 so a_{0} = 18
Then:
a_{1} = 18/2 = 9
a_{2} = ((9-1)/2) + 18 = 22
a_{3}= 22/2 = 11
a_{4} = ((11-1)/2) + 18 = 23
a_{5} = ((23-1)/2) + 18 = 29
a_{6} = ((29-1)/2) + 18 = 32
a_{7} = 32/2 = 16
a_{8} = 16//2 = 8
a_{9} = 8/2 = 4
a_{10} = 4/2 = 2
a_{11} = 2/2 = 1
Once again, the Collatz problem is defined as follows:
Collatz Algorithm: Given any positive integer n, let a_{0} = n. For i > 0, let:
a_{i} = a_{i-1}/2 if a_{i-1} is even
a_{i} = 3*a_{i-1} + 1 if a_{i-1} is odd
Conjecture: Given any positive integer, the above recurrence relation returns 1.
This conjecture seems to hold although it hasn't been proven yet.
While looking at a different problem related to generating powers of 2 backwards, I stumbled upon a Collatz-like algorithm, which I generalize below for all positive integers.
Collatz-like Algorithm #1: Given any positive integer n, let a_{0} = (n+1)/2 if n is odd or a_{0} = n/2 if n is even. For i > 0, let:
a_{i} = a_{i-1}/2 if a_{i-1} is even
a_{i} = (a_{i-1} - 1)/2 + a_{0} if a_{i-1} is odd
Conjecture: Given any positive integer n, the above recurrence relation returns 1, and all integers it reaches before it reaches 1 are strictly less than n
In fact, I claim that:
Claim: If n is even then the Collatz-like Algorithm #1 generates all powers of 2 mod (n-1) in reverse order and if n is odd then it generates all powers of 2 mod n in reverse order.
Example: let n = 36 so a_{0} = 18
Then:
a_{1} = 18/2 = 9
a_{2} = ((9-1)/2) + 18 = 22
a_{3}= 22/2 = 11
a_{4} = ((11-1)/2) + 18 = 23
a_{5} = ((23-1)/2) + 18 = 29
a_{6} = ((29-1)/2) + 18 = 32
a_{7} = 32/2 = 16
a_{8} = 16//2 = 8
a_{9} = 8/2 = 4
a_{10} = 4/2 = 2
a_{11} = 2/2 = 1
Labels:
Collatz Conjecture
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Mathematics
16.3.17
Ghetto Superstar
Last year I visited the world's oldest ghetto, namely the Venetian Ghetto Nuovo. It is the first place in the world where the word ghetto was used to describe a poor segregated community.
The Venetian Ghetto is a part of Venice, Italy where gondolas don't go. That's what I was told when I approached a gondolier asking for directions.
When both Apple maps and Google maps failed to provide a coherent path to my desired destination as well, I had to resort to ancient technology: ask a local who is not in the transportation industry.
A store clerk from a local souvenir shop kindly provided me with the following map, and I somehow managed to find my way.
Venice is by far the most surreal place that I had the opportunity to visit.
Many parts of the city are only accessible by boat, and only emergency service boats are allowed to speed.
According to my slightly cryptic but nevertheless useful map, I had to go to the Grand Canal first and take a waterbus in the opposite direction from where I was initially headed to until reaching the edge of the city.
The Grand Canal runs through the entire length of the Veneto region, which is composed of 117 small islands connected by bridges. The Venetian Ghetto is one of these islands.
I got off near (what looked like) the edge of the city and I went down a small street, then up another small street, and then I somehow ended up on one of the bridges connecting the Venetian Ghetto with the rest of Venice.
At the other side of the bridge there is a tunnel that leads to the main square. The Venetian ghetto was once home to thousands of Sephardic and Ashkenazi Jews, mostly refugees from other parts of Europe where they were unwanted at one point or another.
Although there was a Jewish presence in Venice for centuries prior to the establishment of the Venetian Ghetto, they were not segregated until Venice saw an influx of Jews from Spain after the Alhambra Decree and from other parts of Europe with similar sentiments at the time.
For many of the Sephardic Jews fleeing Spain, Venice was merely a pit stop before reaching the Ottoman Empire.
The Venetian Republic began rounding up and confining Jews to a small island on the edge of the Veneto region on the 29th of March 1516. The island was home to ancient foundries.
There are several synagogues in the ghetto tucked away in common areas of unassuming buildings, and some of them are still in use today. All of them honoured the Venetian Republic in some way with their decor in addition to honouring Ashkenazi, Sephardic, and Italian Jewish tradition.
The word ghetto as we know it today comes from an old Italian word for foundry pronounced as jetto (geto with a soft g as in jet). Apparently, the etymology of the word is a controversial topic but the most plausible theory is that Ashkenazi Jews had trouble pronouncing the soft g so they transformed it into a hard g effectively coining the word ghetto.
Life was not easy in the shanty part of town. Just like in the rest of the world at the time, Jewish people in the Venetian ghetto were only allowed to do the jobs that nobody else liked doing.
They were also not allowed to leave the ghetto at night and they had to wear identifiers that they were Jewish. Living quarters were cramped: large stories inside the buildings were separated into smaller ones to accommodate more people. Entire families were crammed inside single small rooms with low ceilings and sometimes no windows.
Although people who spent time in the ghetto endured many hardships, some of them and many of their descendants thrived and prospered in different parts of the world.
Meanwhile in Spain 500 years after a bill was passed to expel all Jews from the Iberian peninsula, a new bill was introduced to grant citizenship to the descendants of those who were expelled. According to a DNA test I did, it is possible that I'm one such descendant.
The Venetian Ghetto is a part of Venice, Italy where gondolas don't go. That's what I was told when I approached a gondolier asking for directions.
When both Apple maps and Google maps failed to provide a coherent path to my desired destination as well, I had to resort to ancient technology: ask a local who is not in the transportation industry.
A store clerk from a local souvenir shop kindly provided me with the following map, and I somehow managed to find my way.
Venice is by far the most surreal place that I had the opportunity to visit.
An emergency vehicle |
According to my slightly cryptic but nevertheless useful map, I had to go to the Grand Canal first and take a waterbus in the opposite direction from where I was initially headed to until reaching the edge of the city.
A waterbus stop near San Marco Square |
The Grand Canal runs through the entire length of the Veneto region, which is composed of 117 small islands connected by bridges. The Venetian Ghetto is one of these islands.
Rialto Bridge, Venice |
I got off near (what looked like) the edge of the city and I went down a small street, then up another small street, and then I somehow ended up on one of the bridges connecting the Venetian Ghetto with the rest of Venice.
At the other side of the bridge there is a tunnel that leads to the main square. The Venetian ghetto was once home to thousands of Sephardic and Ashkenazi Jews, mostly refugees from other parts of Europe where they were unwanted at one point or another.
Venetian Ghetto Nuovo |
Although there was a Jewish presence in Venice for centuries prior to the establishment of the Venetian Ghetto, they were not segregated until Venice saw an influx of Jews from Spain after the Alhambra Decree and from other parts of Europe with similar sentiments at the time.
For many of the Sephardic Jews fleeing Spain, Venice was merely a pit stop before reaching the Ottoman Empire.
The Venetian Republic began rounding up and confining Jews to a small island on the edge of the Veneto region on the 29th of March 1516. The island was home to ancient foundries.
There are several synagogues in the ghetto tucked away in common areas of unassuming buildings, and some of them are still in use today. All of them honoured the Venetian Republic in some way with their decor in addition to honouring Ashkenazi, Sephardic, and Italian Jewish tradition.
Sephardic Synagogue in the Venetian Ghetto: red and gold were the official colours of the Venetian Republic |
The word ghetto as we know it today comes from an old Italian word for foundry pronounced as jetto (geto with a soft g as in jet). Apparently, the etymology of the word is a controversial topic but the most plausible theory is that Ashkenazi Jews had trouble pronouncing the soft g so they transformed it into a hard g effectively coining the word ghetto.
Life was not easy in the shanty part of town. Just like in the rest of the world at the time, Jewish people in the Venetian ghetto were only allowed to do the jobs that nobody else liked doing.
They were also not allowed to leave the ghetto at night and they had to wear identifiers that they were Jewish. Living quarters were cramped: large stories inside the buildings were separated into smaller ones to accommodate more people. Entire families were crammed inside single small rooms with low ceilings and sometimes no windows.
The view above the main plaza across from Museo Ebraico di Venezia where I learned about the history of the Venetian Ghetto |
Although people who spent time in the ghetto endured many hardships, some of them and many of their descendants thrived and prospered in different parts of the world.
Meanwhile in Spain 500 years after a bill was passed to expel all Jews from the Iberian peninsula, a new bill was introduced to grant citizenship to the descendants of those who were expelled. According to a DNA test I did, it is possible that I'm one such descendant.
Labels:
Traveling
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