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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 $AP3$. 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?

Question B 
How many prime APs beginning with $2$ can you find?

Question C 
How many other arithmetic progressions of prime numbers from the list of primes below can you find?

Next consider some of these three questions:  
Question A 
Why is $3, 5, 7$ the only prime AP with common difference $2$?

Question B 
What is the maximum length of a prime AP with common difference of $6$?

Question C 
If the common difference of a prime AP is $N$ then the maximum length of the prime AP is $N1$.

Now consider some of these three questions:  
Question A 
What is the maxiumum length of a prime AP with common difference $10$?

Question B 
What is the max length of a prime AP with common difference $100, 1000, 10000$ ?

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.

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 