## Thursday, December 31, 2020

### 10/9*8*765/4+321

Happy New Year! (I did my first one ten years ago. My cheat sheet is good to 2101.)

## Friday, December 25, 2020

### 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)

## Thursday, December 24, 2020

### Miracle lights

The scene this morning behind the west fence of Denison Park (Weston). The lights on the left are reflections of the camera flash. On the right, one of the vertical strings of lights (bottoming at 7 o'clock) was actually lit! Naively, I expected to find an extension cord nearby but failed to see one. The path drops somewhat steeply down to the Humber river.

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