Thursday, December 31, 2020

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.

Addendum: Found this twitter post...

Monday, December 14, 2020

Counting digits

1
199981
199982
199983
199984
199985
199986
199987
199988
199989
199990
199991
200001
1599981
1599982
1599983
1599984
1599985
1599986
1599987
1599988
1599989
1599990
1599991
2600001
13199998
13199999
28263827
28263828
35000001
35199981
35199982
35199983
35199984
35199985
35199986
35199987
35199988
35199989
35199990
35199991
35200001
117463825
117463826
242463827
242463828
371599983
371599984
371599985
371599986
371599987
371599988
371599989
371599990
371599991
371599992
371599993
499999984
499999985
499999986
499999987
499999988
499999989
499999990
499999991
499999992
499999993
499999994
500000001
500199981
500199982
500199983
500199984
500199985
500199986
500199987
500199988
500199989
500199990
500199991
500200001
501599981
501599982
501599983
501599984
501599985
501599986
501599987
501599988
501599989
501599990
501599991
502600001
513199998
513199999
528263827
528263828
535000001
535199981
535199982
535199983
535199984
535199985
535199986
535199987
535199988
535199989
535199990
535199991
535200001
1111111110
1111111111
2222222222
3333333333
4444444444
5555555555
6666666666
7777777777
8888888888
9471736170
9471736171
9486799989
9486799990
9486799991
9486799992
9486799993
9486799994
9486799995
9486799996
9486799997
9486799998
9498399989
9498399990
9498399991
9498399992
9498399993
9498399994
9498399995
9498399996
9498399997
9498399998
9628399986
9628399987
9628399988
9628399989
9628399990
9628399991
9628399992
9628399993
9628399994
9628399995
9628399996
9757536170
9757536171
9882536171
9882536172
9971736170
9971736171
9986799989
9986799990
9986799991
9986799992
9986799993
9986799994
9986799995
9986799996
9986799997
9986799998
9998399989
9998399990
9998399991
9998399992
9998399993
9998399994
9998399995
9998399996
9998399997
9998399998
9999999999
10028263827
10028263828
10242463827
10242463828
10371599983
10371599984
10371599985
10371599986
10371599987
10371599988
10371599989
10371599990
10371599991
10371599992
10371599993
10499999984
10499999985
10499999986
10499999987
10499999988
10499999989
10499999990
10499999991
10499999992
10499999993
10499999994
10528263827
10528263828
12222222222
13333333333
14444444444
15555555555
16666666666
17777777777
18888888888
19471736170
19471736171
19486799989
19486799990
19486799991
19486799992
19486799993
19486799994
19486799995
19486799996
19486799997
19486799998
19498399989
19498399990
19498399991
19498399992
19498399993
19498399994
19498399995
19498399996
19498399997
19498399998
19628399986
19628399987
19628399988
19628399989
19628399990
19628399991
19628399992
19628399993
19628399994
19628399995
19628399996
19757536170
19757536171
19882536171
19882536172
19971736170
19971736171
19986799989
19986799990
19986799991
19986799992
19986799993
19986799994
19986799995
19986799996
19986799997
19986799998
19998399989
19998399990
19998399991
19998399992
19998399993
19998399994
19998399995
19998399996
19998399997
19998399998
19999999999
20371599983
20371599984
20371599985
20371599986
20371599987
20371599988
20371599989
20371599990
20371599991
20371599992
20371599993
20499999984
20499999985
20499999986
20499999987
20499999988
20499999989
20499999990
20499999991
20499999992
20499999993
20499999994
23333333333
24444444444
25555555555
26666666666
27777777777
28888888888
29471736170
29471736171
29486799989
29486799990
29486799991
29486799992
29486799993
29486799994
29486799995
29486799996
29486799997
29486799998
29498399989
29498399990
29498399991
29498399992
29498399993
29498399994
29498399995
29498399996
29498399997
29498399998
29628399986
29628399987
29628399988
29628399989
29628399990
29628399991
29628399992
29628399993
29628399994
29628399995
29628399996
29757536170
29757536171
29882536171
29882536172
29971736170
29971736171
29986799989
29986799990
29986799991
29986799992
29986799993
29986799994
29986799995
29986799996
29986799997
29986799998
29998399989
29998399990
29998399991
29998399992
29998399993
29998399994
29998399995
29998399996
29998399997
29998399998
29999999999
30499999984
30499999985
30499999986
30499999987
30499999988
30499999989
30499999990
30499999991
30499999992
30499999993
30499999994
34444444444
35555555555
36666666666
37777777777
38888888888
39471736170
39471736171
39486799989
39486799990
39486799991
39486799992
39486799993
39486799994
39486799995
39486799996
39486799997
39486799998
39498399989
39498399990
39498399991
39498399992
39498399993
39498399994
39498399995
39498399996
39498399997
39498399998
39628399986
39628399987
39628399988
39628399989
39628399990
39628399991
39628399992
39628399993
39628399994
39628399995
39628399996
39757536170
39757536171
39882536171
39882536172
39971736170
39971736171
39986799989
39986799990
39986799991
39986799992
39986799993
39986799994
39986799995
39986799996
39986799997
39986799998
39998399989
39998399990
39998399991
39998399992
39998399993
39998399994
39998399995
39998399996
39998399997
39998399998
39999999999
45555555555
46666666666
47777777777
48888888888
49471736170
49471736171
49486799989
49486799990
49486799991
49486799992
49486799993
49486799994
49486799995
49486799996
49486799997
49486799998
49498399989
49498399990
49498399991
49498399992
49498399993
49498399994
49498399995
49498399996
49498399997
49498399998
49628399986
49628399987
49628399988
49628399989
49628399990
49628399991
49628399992
49628399993
49628399994
49628399995
49628399996
49757536170
49757536171
49882536171
49882536172
49971736170
49971736171
49986799989
49986799990
49986799991
49986799992
49986799993
49986799994
49986799995
49986799996
49986799997
49986799998
49998399989
49998399990
49998399991
49998399992
49998399993
49998399994
49998399995
49998399996
49998399997
49998399998
49999999999
56666666666
57777777777
58888888888
59471736170
59471736171
59486799989
59486799990
59486799991
59486799992
59486799993
59486799994
59486799995
59486799996
59486799997
59486799998
59498399989
59498399990
59498399991
59498399992
59498399993
59498399994
59498399995
59498399996
59498399997
59498399998
59628399986
59628399987
59628399988
59628399989
59628399990
59628399991
59628399992
59628399993
59628399994
59628399995
59628399996
59757536170
59757536171
59882536171
59882536172
59971736170
59971736171
59986799989
59986799990
59986799991
59986799992
59986799993
59986799994
59986799995
59986799996
59986799997
59986799998
59998399989
59998399990
59998399991
59998399992
59998399993
59998399994
59998399995
59998399996
59998399997
59998399998
59999999999
67777777777
68888888888
69471736170
69471736171
69486799989
69486799990
69486799991
69486799992
69486799993
69486799994
69486799995
69486799996
69486799997
69486799998
69498399989
69498399990
69498399991
69498399992
69498399993
69498399994
69498399995
69498399996
69498399997
69498399998
69757536170
69757536171
69882536171
69882536172
69971736170
69971736171
69986799989
69986799990
69986799991
69986799992
69986799993
69986799994
69986799995
69986799996
69986799997
69986799998
69998399989
69998399990
69998399991
69998399992
69998399993
69998399994
69998399995
69998399996
69998399997
69998399998
69999999999
78888888888
79486799989
79486799990
79486799991
79486799992
79486799993
79486799994
79486799995
79486799996
79486799997
79486799998
79498399989
79498399990
79498399991
79498399992
79498399993
79498399994
79498399995
79498399996
79498399997
79498399998
79882536171
79882536172
79986799989
79986799990
79986799991
79986799992
79986799993
79986799994
79986799995
79986799996
79986799997
79986799998
79998399989
79998399990
79998399991
79998399992
79998399993
79998399994
79998399995
79998399996
79998399997
79998399998
79999999999
89999999999
100559404366