Both the two digit numbers n1 and n9 are prime numbers

I just observed for some small $n$ that we can find a permutation of the set $\$ can be permuted in at least one way to obtain a set that has the property that the sum of every two adjacent numbers is a prime number. Can it be that such set exists? Or some known or conjectured fact forbids its existence? $\begingroup$ It's possible at least up to $n=11$. 2 : [1, 2] 3 : [1, 2, 3] 4 : [1, 2, 3, 4] 5 : [1, 4, 3, 2, 5] 6 : [1, 4, 3, 2, 5, 6] 7 : [1, 2, 3, 4, 7, 6, 5] 8 : [1, 2, 3, 4, 7, 6, 5, 8] 9 : [1, 2, 3, 4, 7, 6, 5, 8, 9] 10 : [1, 2, 3, 4, 7, 6, 5, 8, 9, 10] 11 : [1, 2, 3, 4, 7, 10, 9, 8, 5, 6, 11] $\endgroup$ $\begingroup$ @Farewell Suppose $n+1$ and $n+3$ are twin primes. Then placing the numbers in the form $1,n,3,n-2,5,n-4.,...,n-1,2$ does the trick. And most people believe there is an infinite number of twin primes. In fact we might be able to get somewhere using one of the following classes is infinite: twin primes, sexy primes, cousin primes. $\endgroup$ This is still incomplete, but it may help: We say $n$ is prime-permutable if we can permute $\$ so that the sum of adjacent terms is prime. We wish to determine wheter the set $M$ of prime-permutable integers is finite or infinite. ( but if we can it would be great). If it is finite we probably won't be able to prove it since if $n+1$ and $n+3$ are twin primes then $n$ is permutable, via the permutation: $1,n,3,n-2,5,\dots, n-1,2$. The number of suitable permutations for $n\leq 12$ is: $ 2 : 2 $ $ 3 : 2 $ $ 4 : 8 $...

Prime numbers from 1 - 100 Question Can you find all the prime numbers between 1 and 100? Hints Remember, a prime numbers can only be divided by itself and 1. You can use your knowledge of multiples (times tables) to eliminate numbers. Start with the smallest numbers and work your way up. It might help if you draw out a grid so you can spot any patterns. Reveal answer down Here's one way to find all the prime numbers. • 2 is the first prime number. All the other even numbers can be divided by 2, so no other even numbers are prime. • 3 is the next prime number. Work out all the multiples of 3, (6, 9, 12, 15 etc) and forget about these. None of them are primes. • 5 is the next prime. No other multiples of 5 are primes. • Keep going with the other primes (7, 11, 13 etc), removing all the multiples of these as you go. You should end up with a list of 25 prime numbers between 1 and 100. These are shown in the grid below.

If m and n are both two digit numbers and m-n = 11x, is x an integer? The question basically asks whether m-n is a multiple of 11. (1) The tens digit and the units digit of m are same --> m could be: 11, 22, 33, ..., 99 --> m is a multiple of 11. Not sufficiient since no info about n. (2) m+n is a multiple of 11 --> if m=n=11 then the m-n is a multiple of 11 but if m=12 and n=10 then m-n is NOT a multiple of 11. Not sufficient. (1)+(2) From (1) we have that m=. Sufficient. Answer: C. Below might help to understand this concept better. If integers \(a\) and \(b\) are both multiples of some integer \(k>1\) (divisible by \(k\)), then their sum and difference will also be a multiple of \(k\) (divisible by \(k\)): Example: \(a=6\) and \(b=9\), both divisible by 3 ---> \(a+b=15\) and \(a-b=-3\), again both divisible by 3. If out of integers \(a\) and \(b\) one is a multiple of some integer \(k>1\) and another is not, then their sum and difference will NOT be a multiple of \(k\) (divisible by \(k\)): Example: \(a=6\), divisible by 3 and \(b=5\), not divisible by 3 ---> \(a+b=11\) and \(a-b=1\), neither is divisible by 3. If integers \(a\) and \(b\) both are NOT multiples of some integer \(k>1\) (divisible by \(k\)), then their sum and difference may or may not be a multiple of \(k\) (divisible by \(k\)): Example: \(a=5\) and \(b=4\), neither is divisible by 3 ---> \(a+b=9\), is divisible by 3 and \(a-b=1\), is not divisible by 3; OR: \(a=6\) and \(b=3\), neither is divisible by 5...

Related What is a prime number? Prime numbers are natural numbers (positive whole numbers that sometimes include 0 in certain definitions) that are greater than 1, that cannot be formed by multiplying two smaller numbers. An example of a prime number is 7, since it can only be formed by multiplying the numbers 1 and 7. Other examples include 2, 3, 5, 11, etc. Numbers that can be formed with two other natural numbers, that are greater than 1, are called composite numbers. Examples of this include numbers like, 4, 6, 9, etc. Prime numbers are widely used in number theory due to the fundamental theorem of arithmetic. This theorem states that natural numbers greater than 1 are either prime, or can be factored as a product of prime numbers. As an example, the number 60 can be factored into a product of prime numbers as follows: 60 = 5 × 3 × 2 × 2 As can be seen from the example above, there are no composite numbers in the factorization. What is prime factorization? Prime factorization is the decomposition of a composite number into a product of prime numbers. There are many factoring algorithms, some more complicated than others. Trial division: One method for finding the prime factors of a composite number is trial division. Trial division is one of the more basic algorithms, though it is highly tedious. It involves testing each integer by dividing the composite number in question by the integer, and determining if, and how many times, the integer can divide the number evenly....

Prime and Composite Numbers Prime and composite numbers are the two types of numbers, that differ based on the number of factors they have. A prime number is the one that has only two factors and a composite number has more than two factors. A What are Prime Numbers and Composite Numbers? A prime number is a number which has exactly two factors i.e. ‘1’ and the number itself. A composite number has more than two factors, which means apart from getting divided by 1 and the number itself, it can also be divided by at least one positive integer. 1 is not a prime or composite number. Apart from these two, there is also a similar category of numbers which are coprime numbers. A detailed explanation of these numbers is given below. Prime Numbers A prime number is the one which has exactly two factors, which means, it can be divided by only “1” and itself. But “1” is not a prime number. Example of Prime Number 3 is a prime number because 3 can be divided by only two number’s i.e. 1 and 3 itself. • 3/1 = 3 • 3/3 = 1 In the same way, 2, 5, 7, 11, 13, 17 are prime numbers. Composite Numbers A composite number has more than two factors, which means apart from getting divided by the number 1 and itself, it can also be divided by at least one integer or number. We don’t consider ‘1’ as a composite number. Example of Composite Number 12 is a composite number because it can be divided by 1, 2, 3, 4, 6 and 12. So, the number ‘12’ has 6 factors. • 12/1 = 12 • 12/2 =6 • 12/3...

Prime numbers have always been an interesting topic to dive into. However, no one has been able to find a clean and finite formula to generate them. Therefore, mathematicians have relied on algorithms and computational power to do that. Some of these algorithms can be time-consuming while others can be faster. In this tutorial, we’ll go over some of the well-known algorithms to find prime numbers. We’ll start with the most ancient one and end with the most recent one. Most algorithms for finding prime numbers use a method called prime sieves. 2. Sieve of Eratosthenes Sieve of Eratosthenes is one of the oldest and easiest methods for finding prime numbers up to a given number. It is based on marking as composite all the multiples of a prime. To do so, it starts with as the first prime number and marks all of its multiples ( ). Then, it marks the next unmarked number ( ) as prime and crosses out all its multiples ( ). It does the same for all the other numbers up to : However, as we can see, some numbers get crossed several times. In order to avoid it, for each prime , we can start from to mark off its multiples. The reason is that once we get to a prime in the process, all its multiples smaller than have already been crossed out. For example, let’s imagine that we get to . Then, we can see that and have already been marked off by and . As a result, we can begin with . We can write the algorithm in the form of pseudocode as follows: In order to calculate the complexity of th...

A prime number (or prime integer, often simply called a "prime" for short) is a that has no positive integer itself. More concisely, a prime number is a ), making 24 not a prime number. While the term "prime number" commonly refers to prime positive integers, other types of primes are also defined, such as the The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p.31). Although the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909, 1914; Hardy and Wright 1979, p.11; Gardner 1984, pp.86-87; Sloane and Plouffe 1995, p.33; Hardy 1999, p.46), it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own. A good reason not to call 1 a prime number is that if 1 were prime, then the statement of the . In other words, With 1 excluded, the smallest prime is therefore 2. However, since 2 is the only The th prime number is commonly denoted , so , , and so on, and may be computed in the n]. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ... (OEIS The numbers of decimal digits in for , 1, ... is given by 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, ... (OEIS The , represented in the The first few primes are illustrated above as a sequence of binary bits. Euler commented "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it ...

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 Examples:

Ehud de Shalit Professor Ehud de Shalit is a member of the Einstein Institute of Mathematics at the Hebrew University of Jerusalem, specializing in Number Theory. He received his B.Sc. from the Hebrew University and his Ph.D. from Princeton University (1984). In addition to research in mathematics, de Shalit keeps an interest in mathematical education and has delivered many popular lectures in mathematics for general audience. * Age: 12–13 Israel Academy of Sciences and the Arts is a place for curious students who love to learn. In the seventh grade, as part of our elective courses, we are 15 students reading scientific papers that we have chosen. The course is led by Anat Maoz, our junior high school principal, who also holds a Master’s degree in marine biology. Abstract Prime numbers have attracted human attention from the early days of civilization. We explain what they are, why their study excites mathematicians and amateurs alike, and on the way we open a window to the mathematician’s world. From the beginning of human history, prime numbers aroused human curiosity. What are they? Why are the questions related to them so hard? One of the most interesting things about prime numbers is their distribution among the natural numbers. On a small scale, the appearance of prime numbers seems random, but on a large scale there appears to be a pattern, which is still not fully understood. In this short paper, we will try to follow the history of prime numbers since ancient time...