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Today is $03-01-2017$(Date written in English notation) and $3012017$ is a prime number. The other prime number dates, when expanded to include the month and the year as I did for $3^{rd}$ Jan $2017$, falling this month are on $11,12,27$ and $29^{th}$ $\Longrightarrow$ $11012017$, $12012017$,$27012017$ and $29012017$ are prime numbers.

This lead me to a conjecture that in a given year not more than $2$ consecutive dates can be prime. Is there a way I could prove this to be true or false as the case maybe . I dont need a complete proof but some hints that may lead to it.

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    Should I take the date $1-1-1$ as $111$ or $01010001$ – 2017-01-03
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    If three consecutive days are within a month, one of the numbers must be divisble by $3$. This reduces the possible cases drastically – 2017-01-03
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    @WiCK3DPOiSON But we need an example with at least three consecutive "prime-days" – 2017-01-03
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    @WiCK3D POiSON : 1-1-1 should taken as 1011 – 2017-01-03
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    @naveendankal 1011? Don't you want to write the year with 4 digits? – 2017-01-03
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    No the year can be less than or greater than four digits , so the zero is only appended to the months digit and not year or days. – 2017-01-03

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The numbers $$30042119$$ $$1052119$$ and $$2052119$$ correspond to $3$ consecutive "prime-days".

Even $4$ consecutive "prime-days" are possible, for example

$$29061379$$

$$30061379$$

$$1071379$$

$$2071379$$

$5$ consecutive "prime-days" within a year are impossible because at least one of the numbers must be divisble by $3$

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    Your last statement will hold true for any of the leap year for the month of Feb? – 2017-01-03
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    @user366398: yes because three of them must be in the same month and have the same number of digits in the day. – 2017-01-03
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    @RossMillikan Correct, it is not even necessary that the number of digits is the same. The argument also holds if we have the $8th$ , $9th$ and $10th$ of a month, for example – 2017-01-03
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    @Peter: true. I was worried about carries, but as divisibility by $3$ only cares about the sum of the digits it doesn't matter whether you use 08, 09,10 or 8,9,10 you still get it – 2017-01-03