On the product of consecutive primes being the concatenation of consecutive integers (base 2-16)

2*3 = 6 in base 2 is 110: 1~10 = (1,2)

2*3 = 6 in base 4 is 12: 1~2 = (1,2)

3*5 = 15 in base 6 is 23: 2~3 = (2,3)

3*5 = 15 in base 13 is 12: 1~2 = (1,2)

5*7 = 35 in base 2 is 100011: 10~0011 = (2,3)

5*7 = 35 in base 4 is 203: 2~03 = (2,3)

5*7 = 35 in base 16 is 23: 2~3 = (2,3)

23*29 = 667 in base 6 is 3031: 30~31 = (18,19)

43*47 = 2021 in base 10 is 2021: 20~21 = (20,21)

53*59 = 3127 in base 5 is 100002: 1~00002 = (1,2)

59*61 = 3599 in base 2 is 111000001111: 1110~00001111 = (14,15)

59*61 = 3599 in base 4 is 320033: 32~0033 = (14,15)

59*61 = 3599 in base 16 is e0f: e~0f = (14,15)

67*71 = 4757 in base 3 is 20112012: 2011~2012 = (58,59)

67*71 = 4757 in base 9 is 6465: 64~65 = (58,59)

103*107 = 11021 in base 12 is 6465: 64~65 = (76,77)

113*127 = 14351 in base 2 is 11100000001111: 1110~0000001111 = (14,15)

113*127 = 14351 in base 4 is 3200033: 32~00033 = (14,15)

1319*1321 = 1742399 in base 11 is a900aa: a9~00aa = (119,120)

2729*2731 = 7452899 in base 14 is dc00dd: dc~00dd = (194,195)

3359*3361 = 11289599 in base 15 is ed00ee: ed~00ee = (223,224)

3593*3607 = 12959951 in base 3 is 220101102201012: 2201011~02201012 = (1975,1976)

3593*3607 = 12959951 in base 9 is 26342635: 2634~2635 = (1975,1976)

120181*120193 = 14444914933 in base 11 is 6142861429: 61428~61429 = (89691,89692)

189913*189929 = 36069986177 in base 3 is 10110002210001011000222: 1011000221~0001011000222 = (22624,22625)

833363*833377 = 694505556851 in base 13 is 50651050652: 50651~050652 = (143885,143886)

3329233*3329251 = 11083852294483 in base 2 is 10100001010010101001000101000010100101010011: 101000010100101010010~00101000010100101010011 = (1321298,1321299)

4999493*4999507 = 24995000249951 in base 10 is 24995000249951: 249950~00249951 = (249950,249951)

75991159*75991169 = 5774657006074871 in base 5 is 22023343440022023343441: 22023343440~022023343441 = (23652995,23652996)

756395819*756395821 = 572134636513472399 in base 15 is 148e98ed148e98ee: 148e98ed~148e98ee = (223238023,223238024)

5368703993*5368704007 = 28822982639615999951 in base 2 is 11000111111111111110011100000000000000001100011111111111111001111: 1100011111111111111001110~0000000000000001100011111111111111001111 = (26214350,26214351)

5368703993*5368704007 = 28822982639615999951 in base 4 is 120333333303200000001203333333033: 1203333333032~00000001203333333033 = (26214350,26214351)

5368703993*5368704007 = 28822982639615999951 in base 16 is 18fffce00018fffcf: 18fffce~00018fffcf = (26214350,26214351)

Robert Israel and Ed Pegg have posted other very large base-ten examples! I will provide herewith the initial (base 2-16) solutions where the consecutive integers are descending:

2*3 = 6 in base 6 is 10: 1~0 = (1,0)

3*5 = 15 in base 7 is 21: 2~1 = (2,1)

3*5 = 15 in base 15 is 10: 1~0 = (1,0)

5*7 = 35 in base 2 is 100011: 100~011 = (4,3)

5*7 = 35 in base 8 is 43: 4~3 = (4,3)

5*7 = 35 in base 11 is 32: 3~2 = (3,2)

7*11 = 77 in base 5 is 302: 3~02 = (3,2)

7*11 = 77 in base 12 is 65: 6~5 = (6,5)

11*13 = 143 in base 15 is 98: 9~8 = (9,8)

13*17 = 221 in base 6 is 1005: 10~05 = (6,5)

29*31 = 899 in base 7 is 2423: 24~23 = (18,17)

31*37 = 1147 in base 3 is 1120111: 112~0111 = (14,13)

31*37 = 1147 in base 9 is 1514: 15~14 = (14,13)

41*43 = 1763 in base 5 is 24023: 24~023 = (14,13)

67*71 = 4757 in base 11 is 3635: 36~35 = (39,38)

383*389 = 148987 in base 5 is 14231422: 1423~1422 = (238,237)

409*419 = 171371 in base 13 is 60005: 6~0005 = (6,5)

1499*1511 = 2264989 in base 12 is 912911: 912~911 = (1310,1309)

1889*1901 = 3590989 in base 7 is 42344233: 4234~4233 = (1495,1494)

4583*4591 = 21040553 in base 11 is 10971096: 1097~1096 = (1437,1436)

11827*11831 = 139925237 in base 13 is 22cb22ca: 22cb~22ca = (4899,4898)

38011*38039 = 1445900429 in base 6 is 355250355245: 355250~355245 = (30990,30989)

1079783*1079797 = 1165946444051 in base 11 is 40a52540a524: 40a525~40a524 = (658146,658145)

1628329*1628353 = 2651494412137 in base 5 is 321420230032142022: 32142023~0032142022 = (271513,271512)

193746481*193746491 = 37537700837348171 in base 5 is 303330120142303330120141: 303330120142~303330120141 = (153754422,153754421)

1008660187*1008660197 = 1017395382925476839 in base 7 is 1551530341515515303414: 15515303415~15515303414 = (514530735,514530734)

11999987983*11999988017 = 143999712000143999711 in base 10 is 143999712000143999711: 143999712~000143999711 = (143999712,143999711)