Problems & Puzzles: Puzzles

 Puzzle 802. A follow up to Puzzle 801 Now please use the following schemes following scheme: Scheme #1 p*np(p)+2=q q*np(q)+2=r and so on... where np(x) is the next larger prime to x. and the following scheme: Scheme #2 p*np(p)-2=q q*np(q)-2=r and so on... Q. Find the prime p generating the largest stepladder you may produce using both schemes.

Contributions came from Abhiram R. Davesh.

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Abhiram wrote:

Scheme #1
[3, 17]
[19, 439, 194479]
[428473, 183595967299, 33707479215799316047363, 1136194155083542874332193516556999186404864113]
[8896245899, 79143191113266810001, 6263644699591074569835367687614902580019,
39233244922715362793475381385596009951250260660980149317795128782524618801241123, 15392475071657707934535232268831848638874078440156871564447620632049548774729215183447720792/
17193131153374092671931636119347606787768320359598650756404428499313]

Scheme #2
[3, 13]
[14251, 203518529, 41419794088546187]
[1966819, 3868400580587, 14964523052001890698682177, 223936950173895981512027830625106538808532841645451]
[1503571523, 2260727351840826941, 5110888159361238358253906414475765461,
26121177777498906976683473569950649564065051737998596024058678916153506349, 6823159284837027373935181221285069687995286766892362177642044944791684926304088562275153834282099871/
78274244125042028972589633578677762511761995491]

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