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 Emir P. Prime (Posted on 2007-04-19)

Can you identify the five-digit sphenic palindrome

• where the sum of its digits is a palindromic prime,
• where of one its factors is a multi-digit palindromic prime, and
• the sum of its factors is also a palindromic prime?

Can you identify any other sphenic palindrome that has these characteristics?

A sphenic number is a positive integer and product of three distinct primes.
A palindromic number is a number that is the same when written forwards or backwards.

 See The Solution Submitted by Dej Mar Rating: 4.0000 (2 votes)

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 computer solution | Comment 1 of 4

CLS

FOR i = 10000 TO 99999
n\$ = LTRIM\$(STR\$(i))
good = 1
FOR j = 1 TO LEN(n\$)
IF MID\$(n\$, j, 1) <> MID\$(n\$, LEN(n\$) + 1 - j, 1) THEN good = 0: EXIT FOR
NEXT
sod = 0
FOR j = 1 TO 5
sod = sod + VAL(MID\$(n\$, j, 1))
NEXT
n\$ = LTRIM\$(STR\$(sod))
FOR j = 1 TO LEN(n\$)
IF MID\$(n\$, j, 1) <> MID\$(n\$, LEN(n\$) + 1 - j, 1) THEN good = 0: EXIT FOR
NEXT
factor sod, s\$
IF n\$ <> LTRIM\$(s\$) THEN good = 0

IF good THEN
factor i, s\$
s\$ = LTRIM\$(RTRIM\$(s\$))
ix = INSTR(s\$, " ")
IF ix THEN
ix2 = INSTR(ix + 1, s\$, " ")
IF ix2 THEN
ix3 = INSTR(ix2 + 1, s\$, " ")
IF ix3 = 0 THEN
f1 = VAL(LEFT\$(s\$, ix - 1))
f2 = VAL(MID\$(s\$, ix + 1, ix2 - ix - 1))
f3 = VAL(MID\$(s\$, ix2 + 1))
n\$ = LTRIM\$(STR\$(f1 + f2 + f3))
FOR j = 1 TO 2
IF MID\$(n\$, j, 1) <> MID\$(n\$, LEN(n\$) + 1 - j, 1) THEN good = 0: EXIT FOR
NEXT
IF good THEN
factor f1 + f2 + f3, f\$
IF VAL(f\$) = f1 + f2 + f3 THEN
PRINT i, f1; f2; f3
END IF
END IF
END IF
END IF
END IF
END IF
NEXT

SUB factor (num, s\$)
s\$ = "": n = ABS(num): IF n > 0 THEN limit = sqroot(n):  ELSE limit = 0
IF limit <> INT(limit) THEN limit = INT(limit + 1)
dv = 2: GOSUB DivideIt
dv = 3: GOSUB DivideIt
dv = 5: GOSUB DivideIt
dv = 7
DO UNTIL dv > limit
GOSUB DivideIt: dv = dv + 4 '11
GOSUB DivideIt: dv = dv + 2 '13
GOSUB DivideIt: dv = dv + 4 '17
GOSUB DivideIt: dv = dv + 2 '19
GOSUB DivideIt: dv = dv + 4 '23
GOSUB DivideIt: dv = dv + 6 '29
GOSUB DivideIt: dv = dv + 2 '31
GOSUB DivideIt: dv = dv + 6 '37
IF INKEY\$ = CHR\$(27) THEN s\$ = CHR\$(27): EXIT SUB
LOOP
IF n > 1 THEN s\$ = s\$ + STR\$(n)
EXIT SUB

DivideIt:
DO
q = INT(n / dv)
IF q * dv = n AND n > 0 THEN
n = q: s\$ = s\$ + STR\$(dv): IF n > 0 THEN limit = sqroot(n):  ELSE limit = 0
IF limit <> INT(limit) THEN limit = INT(limit + 1)
ELSE
EXIT DO
END IF
LOOP
RETURN
END SUB

finds 32123, with prime factors 7, 13, 353, which add to 373, a prime. The sum of digits of 32123 is 11.

 Posted by Charlie on 2007-04-19 12:05:26

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