Forming Large Primes

 I was surprised to come across the tweet shown in Figure 1 from Fermat's Library:

Figure 1: permalink

While this is surprising enough, it's also interesting that, whether we are counting down from 82 or counting up from 1, none of the intermediate numbers is prime. The prime has 155 digits. However, if the digits are removed or added one by one, then there are two intermediate primes (permalink). These are:

746454443424140393837363534333231302928272625242322212019181716151413121110987654321

21110987654321

If we start adding the numbers to the right of the 1 instead of to the left, it has been shown that no primes exist up to \(n\)=344869 (see comments to OEIS A007908). The search for such a prime formed in this manner is continuing. 

This got me thinking about other ways to create large primes, starting from 1 each time. What if we add only prime numbers to the left or right. How long before a prime is created? 

ADDING ONLY PRIMES TO THE LEFT

Well, let's starting adding primes to the left of 1 up to a limit of 1000. This what I found (permalink):

231917131175321 with 15 digits

41373129231917131175321 with 23 digits

898379737167615953474341373129231917131175321 with 45 digits

ADDING ONLY PRIMES TO THE RIGHT

What if the primes are added to the right of the 1? This is the result (permalink):

1235711 with 7 digits

123571113171923 with 15 digits

ADDING ONLY ODD NUMBERS TO THE LEFT

What if we add only odd numbers to the left of the 1? This is the result (permalink):

31 with 2 digits

737169676563615957555351494745434139373533312927252321191715131197531 with 69 digits

12312111911711511311110910710510310199979593918987858381797775737169676563615957555351494745434139373533312927252321191715131197531 with 131 digits

ADDING ONLY ODD NUMBERS TO THE RIGHT

What if we add the odd numbers to the right of the 1? Here are the results (permalink):

13 with 2 digits

135791113151719 with 15 digits

135791113151719212325272931 with 27 digits

135791113151719212325272931333537394143454749515355575961636567 with 63 digits

135791113151719212325272931333537394143454749515355575961636567697173757779818385878991939597 with 93 digits

This investigation is never-ending so I'll stop there but counting down from 82 to 1 still holds the record for the longest prime at 155 digits. The permalinks above can be easily modified to investigate other variations.