if we are following some other base other than 10, say 8 and we represent a number its value being 43 ( if this were a decimal then the number would be prime ).  but when we convert it in decimals ( as it were not a decimal in the first place ) its value is 35. now the problem arises to me that if we change the base of numbers even primes are no longer primes and the so called special numbers lose their specialty.
121918
we all know that primes are special numbers, but let me ask you one thing that 'if we change the base of our widely used decimal to something else, lets say to octal, would the distribution of primes also change; or it is irrespective of the number system we use ?'  also if more basically we adopt binary number system which numbers would we call primes and how would sieve work ?  ask me if you are still not getting my question
113587
Hello everyone this is my first question here so if there are any suggested edits  feel free to participate! Is it true that for every $n$ there two prime numbers $p$ and $q$ with $n\mid p-q$? I have verified this empirically but  am not sure if there is a proof for this
122701
reports:  Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers [...]. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them. Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9.  Do these insights make breaking keys based on primes more mathematically plausible in a shorter amount of time?
108191
I was looking at a 10x10 multiplication table, and I decided to count the unique products. There are 42 out of a possible 100 numbers represented. I had to wonder, why 42? I counted the 58 non-listed numbers -- most of them are either primes >10 or multiples of those primes. I couldn't figure out a satisfying reason like that for 75, 84, 96 and 98. I feel like the number 42 should be calculable from the natural log of 100 and 10, or something, but I can't really figure it out.  Also, I counted the frequency of all the numbers represented. They all have either 1, 2, 3, or 4 instances. The six numbers that have only 1 instance are all squares -- those of 1, 5, 7, 8, 9, and 10. This makes sense -- squares don't fall into that 4x5 = 5x4 redundancy. The 23 numbers with 2 instances are kind of the default I guess. The four with 3 are the remaining squares -- 4, 9, 16, and 36. 4 and 9 make sense to me, as they are squares within the basic range of the times table, so they are hit by themselves, their square root, and 1. I'm having trouble accounting for why 16 and 36 have three representatives. I'm also having trouble seeing the pattern (or patterns) in the numbers with four representatives.  I ultimately want to find a way to, given a times table of any size, calculate whether a given number will appear, and how many times.  Also, I'm pretty sure that for times tables 7x7 and below, more than half of the possible numbers are represented on the table, but for times tables above 7x7, less than half are represented. Why is 7 the turning point?
41829
Most people use base 10; it's obviously the common notation in the modern world.  However, if we could change what became the common notation, would there be a better choice?  I'm aware that it very well may be that there is no intrinsically superior base, but for the purposes of humans, is there a better one?  I've heard from sources such as  and  that base 12 is better, from  that base 8 is better, and, being into computer science, I would say that base 16 is the most handy.  Base 12 does seem to be the most supported non-base 10 number system, mainly due to the following reason pointed out by George Dvorsky:     First and foremost, 12 is a highly composite number — the smallest   number with exactly four divisors: 2, 3, 4, and 6 (six if you count 1   and 12). As noted, 10 has only two. Consequently, 12 is much more   practical when using fractions — it's easier to divide units of   weights and measures into 12 parts, namely halves, thirds, and   quarters.   And, on top of that, previous societies considered very advanced used other systems, such as the Mayans using base 20, and the Babylonians using base 60.  So, summarized, my question is: Is there an intrinsically superior base? If not, is there one that would be best for society's purposes? Or does the best base depend on the context it is being used in?
41147
Problem :   Solve in $\mathbb N$ :   $8n+9\equiv 0\pmod{1163}$  The mathematics give me : $n=435+1163k$ , $k\in\mathbb N$   $1163$ is a prime number   $8n\equiv -9\pmod{1163}$   $8n\equiv 1154\pmod{1163}$   I don't know any ideas to complete my work ?   I have to see your hints
104546
My question is about number theory about the prime numbers.  Here is the question:  Prove that there is only one prime trio of the form $p$,$p+2$,$p+4$.  I mean,we can only find the primes 3,5,7  That satisfies the condition.
73300
Possible Duplicate:          This is not a practical problem. I am asking out of curiosity.  Any links/references are most welcome.  Say, we write the digits of $\pi$ in base $10$.  Does this sequence of digits contain every possible finite length digit sequence?  What about $e$, $\sqrt{2}$ or some other commonly known irrational numbers?  Is this property of numbers independent of base?  If a number has this property when written in base $10$, will it also have it in base $2$, $3$ and all other bases?
111147
Prove that there is no arithmetic progression that consists only of primes.  A question that I've been set; I'm guessing it makes use of primes being written in the form 4k+1 and 4k+3? Not sure where to start.  Thanks.
38510
Prove: If a prime number $p\in \mathbb N$ is from the form $p=4k+3,k\in \mathbb N$, then its also a prime number in $\mathbb Z[i]$,i.e. if $p|(z_1\cdot z_2)$ then $p|z_1$ or $p|z_2$.  I dont have any idea how to solve it, so I am looking for some hints.  Thanks in advance :)
41284
Define the sequence $P(n)=\sum_{i=1}^{n}p_i$, where $p_i$ is the $i$-th prime number.  I observed for some small $n$ that sometimes this sum evaluates to a prime number, for example $P(2)=2+3=5$ and $P(4)=2+3+5+7=17$ and $P(6)=2+3+5+7+11+13=41$.  So it is natural to ask:     Is it true that there is a sequence of natural numbers $\{n_i:i \in \mathbb N\}$ such that all numbers in the set $\{P(n_i):i \in \mathbb N\}$ are prime numbers?
94491
This .  Why is that number recommended and what are the theoretical and practical impacts &amp; risks of making that number higher or lower?   What are the impacts of making that value a non-Fermat number, or simply non prime?
45064
31,331,3331, 33331,333331,3333331,33333331 are prime. This law can continue it? Will there emerge a composite number? Without using a computer how to judge.
51712
I was going through the fundamental theorem in Number Theory where any non zero integer n can be represented as a product of distinct primes. A related problem with this theorem is to prove that for every such number, there exists a prime $p$ such that $p&lt; \sqrt n$. I was wondering if there is any mathematical proof that no prime $p$ exists for the number $n$ such that $p&gt; \sqrt n$.
37368
So recently, an absurdly large prime number was found, but a lot of prime numbers less than it are still not known. I am wondering up to where we know all the primes.  I put "currently publicly known" because there is a chance that some government agency has a longer list for crypto reasons or something like that.
16284
Given a number presented in decimal form $n=\sum_{k=0} d_k10^k$ there are some nice thumb rules to decide whether it is divisible with a prime number  $p\in\{7,11,13,17,19,23,29\}$.   For each of this numbers there is a number  $a_p$ such that $p|n\iff p|(a_p\cdot d_0+\sum_{k=0} d_{k+1}10^k)$.  $a_7=-2,\;a_{11}=-1,\;a_{13}=4,\;a_{17}=-5,\;a_{19}=2,\;a_{23}=7,\;a_{29}=3$  E.g. $\;19\cdot 17=323$ and $32+2\cdot 3=38=2\cdot 19$ and $32-5\cdot 3=17$.  Is there a number $a_{31}$ that works the same way? Or $a_p,\;p\ge37$?
63318
I was reading Wikipedia, and it was given that "all primes are of the form 6k ± 1"  (other than 2 and 3), where k = 1,2,3,4,...  Is this statement correct? If yes, can we use this to test if a given number is a prime number? For instance, we can say that 41 is a prime number, since there exists an integer K (K >= 1), such that 6k - 1 = 41 ==> k = 7.  I am confused why this test cannot be used to test if a number is prime?  Thanks, Sekhar