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I would like to use Zapf's Palatino font for the text in my document, which also includes a lot of maths. And I would like a good font for the maths. This is a snapshot of what I am doing.

documentclass[10pt,letterpaper]memoir
usepackageamsmath
usepackageamssymb
usepackagemathpple
usepackageupgreek
usepackagemathpazo
usepackagelipsum

begindocument

chapterDemo

Some regular text. lipsum[1]

Now for some maths.

begindisplaymath
P_N-1 + m = fracCN-1 + m binomN_2 - N_1m
 alpha^mbeta^(N_2 - N_1) -m
enddisplaymath

The number 27 is the sum of the first 5 odd prime numbers: $27 = 1+3+5+7+11$

enddocument

To me the text and math fonts seem a good fit but the text does not look like Palatino.

I suggest that you not load the mathpple, mathpazo, and upgreek packages and, instead, load the newpxtext and newpxmath packages.

documentclass[10pt,letterpaper]memoir
usepackageamsmath,amssymb
%usepackagemathpple
%usepackageupgreek
%usepackagemathpazo
usepackagelipsum
usepackagenewpxtext,newpxmath

begindocument

Some regular text. lipsum[2]

Now for some maths.
begindisplaymath
P_N-1 + m = fracCN-1 + m binomN_2 - N_1m,
 alpha^mbeta^(N_2 - N_1) -m
enddisplaymath
The number 27 is the sum of the first 5 odd prime numbers: $27 = 1+3+5+7+11$.

$alphabetagammadelta$ vs. $upalphaupbetaupgammaupdelta$

enddocument

You can also do this with OpenType fonts using unicode-math.

documentclass[10pt,letterpaper]memoir
usepackageamsmath
usepackageunicode-math
usepackagemicrotype
usepackagelipsum

defaultfontfeaturesScale=MatchLowercase
setmainfontTeX Gyre Pagella
setmathfontAsana Math

begindocument

chapterDemo

Some regular text. lipsum[1]

Now for some maths.

begindisplaymath
P_N-1 + m = fracCN-1 + m binomN_2 - N_1m
 alpha^mbeta^(N_2 - N_1) -m
enddisplaymath

The number 27 is the sum of the first 5 odd prime numbers: $27 = 1+3+5+7+11$

enddocument

With LuaLaTeX, including microtype will get you both font protrusion and font expansion, which cuts down on the amount of hyphenation substantially.

You can also substitute Palatino or Palatino Linotype as the main font if you have it, or TeX Gyre Pagella Math as the math font. A sans-serif font that matches well is Hermann Zapf’s Optima, and you might mix in some of Zapf’s Neo Euler in math mode (I like its upright letters and its script alphabet).