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| stats | profile views | 13,671 |
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May 12 |
comment |
If all of $p, p + 2, p + 6$ and $p + 8$ are prime, then $p \equiv k \pmod d$ Take $p>5$. We cannot have $p\equiv 0\pmod 5$. We cannot have $p\equiv 2\pmod 5$, else $p+8$ divisible by $5$, so not prime. We cannot have $p\equiv 3\pmod{5}$, else $p+2$ not prime. Cannot have $p\equiv 4\pmod{5}$, else $p+6$ not prime. Leaves $p\equiv 1\pmod{5}$ as only possibility. Work similarly mod $6$. We cannot have $p\equiv 1\pmod{6}$, else $p+2$ is divisible by $3$. So $p\equiv -1\pmod{6}$. From $p\equiv 1\pmod{5}$ and $p\equiv -1\pmod{6}$ we conclude $p\equiv{11}\pmod{30}$. For that, easiest is by working mod $30$. (more) |
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May 12 |
comment |
How to find the factors of numbers around 1e7? @AlexBecker: A comment is only a comment, something off-hand. |
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May 12 |
comment |
How to find the factors of numbers around 1e7? Success in Project Euler problems involves good use of relevant mathematics, and usually very clever programming, with the emphasis on the latter. I am not sure what you are asking. If you know the prime power factorization $n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$ of $n$, you can get all factors of $n$ by multiplying together $0$ to $a_1$ $p_1$'s with $0$ to $a_2$ $p_2$'s, and so on. The total number of factors is $(a_1+1)(a_2+1)\cdots(a_k+1)$. |
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May 12 |
revised |
Number of combinatorial progressions minor typo fix, plus made the range of summation explicit |
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May 12 |
comment |
Number of combinatorial progressions @Shahab: Thanks for spotting the typo. Indeed it is $m2^{m-1}$. And yes, we get, in my notation, $N2^m -(3m)2^{m-1}$, which, in terms of $k$, is what you wrote above. |
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May 12 |
revised |
Number of combinatorial progressions added 1 characters in body |
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May 12 |
revised |
Number of combinatorial progressions added 32 characters in body |
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May 12 |
revised |
Number of combinatorial progressions added 46 characters in body |
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May 12 |
revised |
Number of combinatorial progressions added 46 characters in body |
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May 12 |
answered | Number of combinatorial progressions |
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May 11 |
revised |
Does a random action have probability? added 160 characters in body |
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May 11 |
revised |
If all of $p, p + 2, p + 6$ and $p + 8$ are prime, then $p \equiv k \pmod d$ added 33 characters in body |
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May 11 |
revised |
Does a random action have probability? added 34 characters in body |
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May 11 |
answered | Does a random action have probability? |
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May 11 |
answered | If all of $p, p + 2, p + 6$ and $p + 8$ are prime, then $p \equiv k \pmod d$ |
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May 11 |
answered | Is it possible to construct this triangle? |
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May 11 |
revised |
Mutual Uniqueness of Operations in PA models added 106 characters in body |
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May 11 |
answered | Mutual Uniqueness of Operations in PA models |
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May 11 |
revised |
$\sum^n_{j=0} (-1)^{j-1}j $ deleted 86 characters in body |
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May 11 |
answered | $\sum^n_{j=0} (-1)^{j-1}j $ |
