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A simple arithmetic progression of three primes starts at $3$ with common difference $4$, giving rise to the progression of prime numbers $$ 3, 7,11 $$ This is an example of $AP-3$. Note that the sequence stops here because $11+4=15$, which is not a prime number. Another short arithmetic progression starts at $7$ with common difference $6$ $$ 7, 13, 19 $$ |
Consider some of these three questions first: | |
Question A |
Can you find an arithmetic progression of four primes?
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Question B |
How many prime APs beginning with $2$ can you find?
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Question C |
How many other arithmetic progressions of prime numbers from the list of primes below can you find?
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Next consider some of these three questions: | |
Question A |
Why is $3, 5, 7$ the only prime AP with common difference $2$?
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Question B |
What is the maximum length of a prime AP with common difference of $6$?
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Question C |
If the common difference of a prime AP is $N$ then the maximum length of the prime AP is $N-1$.
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Now consider some of these three questions: | |
Question A |
What is the maxiumum length of a prime AP with common difference $10$?
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Question B |
What is the max length of a prime AP with common difference $100, 1000, 10000$ ?
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Question C |
What are the possible lengths of prime APs with common difference $2p$, where $p$ is prime? Consider $p=3$ and $p> 3$ separately.
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Prove that if an AP-$k$ does not begin with the prime $k$, then the common difference is a multiple of the primorial $k$#$ = 2\cdot 3\cdot 5 \cdot \dots \cdot j$, where $j$ is the largest prime not greater than $k$. |
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 | 1009 | 1013 | 1019 | 1021 | 1031 | 1033 | 1039 | 1049 | 1051 | 1061 | 1063 | 1069 |